Is migration between Varna really allowed in Hinduism? If yes, please give examples along with reference.
Yes Varna migration is possible.
Change of Varna
Narada said, 'If in members born in a certain Varna the qualities pertaining to another Varna are seen, they (the former) are to be classified as belonging to the latter Varna.'
Srimad Bhagavata Purana VII.11.35
Sage Pulastya's views on how to become a Brahmana
Sage Pulastya said,"..by vows, by investiture of the sacred, by fasts, by rites, and by Mantras, one becometh a Brahmana."
(Mahabharata, Aranya Parva LXXXIII)
I am supplementing my answer with actual examples of persons who have changed Varna.
Examples of Change of Varna in Hindu scripture
Bhagavan Rishabha, realising that the region of his advent was a place dominated by Vedic rituals, adopted the life of a religious student under a teacher with gifts, came back home with his blessings. He adopted the householder’s station of life in order to teach the world about the duties of that order, observed all the ceremonials and duties laid down in the scriptures, married a girl named Jayanti given to him by Indra, and begot by her a hundred sons equal to himself in all respects. Of all these sons, Bharata was the eldest and noblest. This Ajanabha Varsha came to be known after him as Bharatavarsha. Next to him, the eldest nine other sons ….. were elder to the remaining ninety. Among these ninety, another nine … became great devotees and teachers of the Bhakti cult. ….. The remaining eighty one of the brothers, who were humble in nature, learned in the Vedas, adepts in sacrificial rites, and extremely pure through their observances, became Brahmanas according to their father’s instruction.
Srimad Bhagavata Purana, V.4.8-13
At Kamyaka, Kusika’s son had quaffed the Soma juice with Indra. Then abandoning the Kshatriya order, he began to say,’‘I am a Brahmana’.
Mahabharata, Vana Parva, Section LXXXVII
I don't know whether Varna migration is really allowed in Hinduism or not, but one such example of Varna migration is mentioned in Mahabharata Anusasana Parva chapter 30. The story explains how a Kshatriya king named Vitahavya obtained the status of a Brahmana.
Yudhishthira said, 'I have heard this great narrative, O perpetuator of Kuru's race. Thou, O foremost of eloquent men, hast said that the status of a Brahmana is exceedingly difficult of acquisition. It is heard, however, that in former times the status of a Brahmana had been acquired by Viswamitra. Thou, however, O best of men, tellest us that status is incapable of being acquired. I have also heard that king Vitahavya in ancient times succeeded in obtaining the status of a Brahmana. I desire to hear, O puissant son of Ganga, the story of Vitahavya's promotion. By what acts did that best of kings succeed in acquiring the status of a Brahmana? Was it through some boon (obtained from some one of great puissance) or was it through the virtue of penances? It behoveth thee to tell me everything.'
"Bhishma said, 'Hear, O monarch, how the royal sage Vitahavya of great celebrity succeeded in ancient times in acquiring the status of a Brahmana that is so difficult to attain and that is held in such high reverence by all the world. While the high-souled Manu in days of yore was employed in righteously ruling his subjects, he obtained a son of righteous soul who became celebrated under the name of Saryati. In Saryati's, race, O monarch, two kings took their birth, viz., Haihaya and Talajangha. Both of them were sons of Vatsa, O foremost of victorious kings. Haihaya, O monarch, had ten wives. Upon them he begot, O Bharata, a century of sons all of whom were highly inclined to fighting. All of them resembled one another in features and prowess. All of them were endued with great strength and all of them were possessed of great skill in battle. They all studied the Vedas and the science of weapons thoroughly. In Kasi also, O monarch, there was a king who was the grandfather of Divodasa. The foremost of victorious men, he was known by the name of Haryyaswa. The sons of king Haihaya, O chief of men (who was otherwise known by the name of Vitahavya), invaded the kingdom of Kasi and advancing to the country that lies between the rivers Ganga and Yamuna, fought a battle with king Haryyaswa and also slew him in it. Having slain king Haryyaswa in this way, the sons of Haihaya, those great car-warriors, fearlessly went back to their own delightful city in the country of the Vatsas. Meanwhile Haryyaswa's son Sudeva, who looked like a deity in splendour and who was a second god of righteousness, was installed on the throne of Kasi as its ruler. The delighter of Kasi, that righteous-souled prince ruled his kingdom for sometime, when the hundred sons of Vitahavya once more invaded his dominions and defeated him in battle. Having vanquished king Sudeva thus, the victors returned to their own city. After that Divodasa, the son of Sudeva, became installed on the throne of Kasi as its ruler. Realising the prowess of those high-souled princes, viz., the sons of Vitahavya, king Divodasa, endued with great energy, rebuilt and fortified the city of Baranasi at the command of Indra.
The territories of Divodasa were full of Brahmanas and Kshatriyas, and abounded with Vaisyas and Sudras. And they teemed with articles and provisions of every kind, and were adorned with shops and marts swelling with prosperity. Those territories, O best of kings, stretched northwards from the banks of Ganga to the southern banks of Gomati, and resembled a second Amravati (the city of Indra). The Haihayas once again, O Bharata, attacked that tiger among kings, as he ruled his kingdom. The mighty king Divodasa endued with great splendour, issuing out of his capital, gave them battle. The engagement between the two parties proved so fierce as to resemble the encounter in days of old between the deities and the Asuras. King Divodasa fought the enemy for a thousand days at the end of which, having lost a number of followers and animals, he became exceedingly distressed. King Divodasa, O monarch, having lost his army and seeing his treasury exhausted, left his capital and fled away. Repairing to the delightful retreat of Bhardwaja endued with great wisdom the king, O chastiser of foes joining his hands in reverence, sought the Rishi's protection. Beholding King Divodasa before him, the eldest son of Vrihaspati, viz., Bharadwaja of excellent conduct, who was the monarch's priest, said unto him, What is the reason of thy coming here? Tell me everything, O king. I shall do that which is agreeable to thee, without any scruple.'
"The king said, 'O holy one, the sons of Vitahavya have slain all the children and men of my house. I only have escaped with life, totally discomfited by the foe. I seek thy protection. It behoveth thee, O holy one, to protect me with such affection as thou hast for a disciple. Those princes of sinful deeds have slaughtered my whole race, leaving myself only alive.'
"Bhishma continued, 'Unto him who pleaded so piteously, Bharadwaja of great energy said, Do not fear! Do not fear! O son of Sudeva, let thy fears be dispelled. I shall perform a sacrifice, O monarch, in order that thou mayst have a son through whom thou shalt be able to smite thousands upon thousands of Vitahavya's party. After this, the Rishi performed a sacrifice with the object of bestowing a son on Divodasa. As the result thereof, unto Divodasa was born a son named Pratarddana. Immediately on his birth he grew up like a boy of full three and ten years and quickly mastered the entire Vedas and the whole of arms. Aided by his Yoga powers, Bharadwaja of great intelligence had entered into the prince. Indeed, collecting all the energy that occurs in the object of the universe, Bharadwaja put them together in the body of prince Pratarddana. Put on shining mail on his person and armed with the bow, Pratarddana, his praises sung by bards and the celestial Rishis, shone resplendent like the risen star of day. Mounted on his car and with the scimitar tied to his belt, he shone like a blazing fire. With scimitar and shield and whirling his shield as he went, he proceeded to the presence of his sire. Beholding the prince, the son of Sudeva, viz., king Divodasa, became filled with joy. Indeed, the old king thought the sons of his enemy Vitahavya as already slain. Divodasa then installed his son Pratarddana as Yuvaraja, and regarding himself crowned with success became exceedingly happy. After this, the old king commanded that chastiser of foes, viz., prince Pratarddana to march against the sons of Vitahavya and slay them in battle. Endued with great powers. Pratarddana, that subjugator of hostile cities speedily crossed Ganga on his car and proceeded against the city of the Vitahavyas. Hearing the clatter produced by the wheels of his car, the sons of Vitahavya, riding on their own cars that looked like fortified citadels and that were capable of destroying hostile vehicles, issued out of their city. Issuing out of their capital, those tigers among men, viz., the sons of Vitahavya, who were all skilful warriors cased in mail, rushed with uplifted weapons towards Pratarddana, covering him with showers of arrows. Encompassing him with innumerable cars, O Yudhisthira, the Vitahavyas poured upon Pratarddana showers of weapons of various kinds like clouds pouring torrents of rain on the breast of Himavat. Baffling their weapons with his own, prince Pratarddana endued with mighty energy slew them all with his shafts that resembled the lighting fire of Indra. Their heads struck off, O king, with hundreds and thousands of broad-headed arrows, the warriors of Vitahavya fell down with blood-dyed bodies like Kinsuka trees felled by woodmen with their axes on every side. After all his warriors and sons had fallen in battle, king Vitahavya fled away from his capital to the retreat of Bhrigu. Indeed, arrived there, the royal fugitive sought the protection of Bhrigu. The Rishi Bhrigu, O monarch, assured the defeated king of his protection. Pratarddana followed in the footsteps of Vitahavya. Arrived at the Rishi's retreat, the son of Divodasa said in a loud voice.--Ho, listen ye disciples of the high souled Bhrigu that may happen to be present, I wish to see the sage. Go and inform him of this. Recognising that it was Pratarddana who had come, the Rishi Bhrigu himself came out of his retreat and worshiped that best of kings according to due rites. Addressing him then, the Rishi said, Tell me, O king, what is thy business. The king, at this, informed the Rishi of the reason of his presence.'
"The king said, 'King Vitahavya has come here, O Brahmana. Do thou give him up. His sons, O Brahmana, had destroyed my race. They had laid waste the territories and the wealth of the kingdom of Kasi. Hundred sons, however, of this king proud of his might, have all been slain by me. By slaying that king himself I shall today pay off the debt I owe to my father. Unto him that foremost of righteous men, viz., the Rishi Bhrigu, penetrated with compassion, replied by saying, There is no Kshatriya in this retreat. They that are here are all Brahmanas. Hearing these words of Bhrigu that must accord he thought with truth, Pratarddana touched the Rishi's feet slowly and, filled with delight, said, By this, O holy one, I am without doubt, crowned with success, since this king becomes abandoned by the very order of his birth in consequence of my prowess. Give me thy permission, O Brahmana, to leave thee, and let me solicit thee to pray for my welfare. This king, O founder of the race that goes by the name, has been compelled to leave of the very community of his birth, in consequence of my might. Dismissed by the Rishi Bhrigu, king Pratarddana then departed from that retreat, having even as a snake vomits forth its real poison and repaired to the place he had come from. Meanwhile, king Vitahavya attained to the status of a Brahmana sage by virtue of the words only of Bhrigu. And he acquired also a complete mastery over all the Vedas through the same cause.
- Bhishma narrated the story of Vitahavya, who a Kshatriya then later obtained a status of a Brahmana.
- In the race of Saryati, a king named Haihaya (who was otherwise called as Vitahavya), took birth. He had ten wives from whom he begot hundred sons.
- At the time, Kasi was ruled by a king named Haryyaswa. The sons of Vitahvya invaded kasi and defeated him. After this, Haryyaswa's son Sudeva became king of Kasi. Thereafter, even Sudeva was defeated by those hundred sons. After this, the grandson of Haryyaswa, named Divodasa ruled Kasi.
- The sons of Vitahavya once more invaded Kasi and killed all of his army and plundered the treasuries of Kasi. Because of this, Divodasa became very sad and went to the hermitage of Bharadwaja.
- Bharadwaja upon hearing this incident, performed a sacrifice with the object of bestowing a son on Divodasa. Because of which he bestowed a son, named Pratarddana. When Pratarddana born, he resembled like a 13 years old boy.
- Then Bharadwaja collected all the energies of the universe and put them into the body of Pratarddana. Then Pratarddana marched against the sons of Vitahavya, and killed them.
- After the sons of Vitahavya were killed in the battle, he sought refuge in the hermitage of Rishi Bhrigu. Pratarddana also went to Bhrigu after knowing that Vitahavya went there. Pratarddana said to Bhrigu that the sons of Vitahavya destroyed his race, so that only he killed the sons. He also said he wanted to kill Vitahavya too.
- Bhrigu said, there is no need to kill Vitahavya, as he was a Brahmana. Then king Pratarddana said to Bhrigu that as Vitahavya abandoned his very birth order (i.e., Kshatriya Varna), he crowned with success in battle with Vitahavya. Thereafter he left Bhrigu's place.
- Thus, simply by the words of sage Bhrigu, king Vitahavya became a Brahmana along with his descendants. (He may begot sons after this incident).
Is migration between Varna really allowed in Hinduism ?
Yes , the migration between Varnas are allowed in Hinduism and the topic is discussed in Mahabharata - Book 13: Anusasana Parva SECTION CXLIII through conversation between Devi Uma & Lord Mahadeva.
Here are the excerpts-:
The boon-giving Brahma, while he created all creatures, himself said that the distribution of human beings into the four orders dependent on birth is only for purposes of classification.
By these good acts, again, O goddess, when performed, a Sudra becomes a Brahmana, and a Vaisya becomes a Kshatriya.
A pious Kshatriya, by his own acts, becomes a Brahmana. It is with the aid of these acts, O goddess, that a person who has sprung from a degraded order, viz., a Sudra, may become a Brahmana refined of all stains and possessed of Vedic lore,
Even a Sudra, O goddess, that has purified his soul by pure deeds and that has subjugated all his senses, deserves to be waited upon and served with reverence as a Brahmana.
Sudra, if he is established on good conduct, is regarded as possessed of the status of a Brahmana.
Footnote -: 304.1
The sense seems to be this: a Vaisya ultimately becomes a Brahmans by observing the duties indicated in verses 30 to 33. As the immediate reward, however, of his observance of these duties, he becomes a great Kshatriya. What he should next do in order to become a Brahmana is said in the verses that follow.
Although this section is mainly talking about the birth one gets by following his own dharma / duties in next life , the points mentioned above hints us towards allowance of varna migration in Hinduism.
Swami Vivekananda writes (Complete Works, V4, pp 469-70, and available here under the heading Translation: Prose, sub-heading Modern India - http://cwsv.belurmath.org/volume_4/vol_4_frame.htm):
In spite of the spread of education in the West, there is a great hindrance in the way of the rising of the Shudra class, and that is the recognition of caste as determined by the inherence of more or less good or bad qualities. By this very qualitative caste system which obtained in India in ancient days, the Shudra class was kept down, bound hand and foot. In the first place, scarcely any opportunity was given to the Shudra for the accumulation of wealth or the earning of proper knowledge and education; to add to this disadvantage, if ever a man of extraordinary parts and genius were born of the Shudra class, the influential higher sections of the society forthwith showered titular honours on him and lifted him up to their own circle. His wealth and the power of his wisdom were employed for the benefit of an alien caste — and his own caste-people reaped no benefits of his attainments; and not only so, the good-for-nothing people, the scum and refuse of the higher castes, were cast off and thrown into the Shudra class to swell their number. Vasishtha, Nârada, Satyakâma Jâbâla, Vyâsa, Kripa, Drona, Karna, and others of questionable parentage* [footnote 1] were raised to the position of a Brahmin or a Kshatriya, in virtue of their superior learning or valour; but it remains to be seen how the prostitute, maidservant, fisherman, or the charioteer* [footnote 2] class was benefited by these upliftings. Again, on the other hand, the fallen from the Brahmin, the Kshatriya, or the Vaishya class were always brought down to fill the ranks of the Shudras.
*footnote 1: (1) Vasishtha's father was Brahmâ and mother unknown. (2) Narada's mother was a maidservant and father unknown. (3) Satyakama Jabala's mother was a maidservant, by name Jabâlâ, and father unknown. (4) Vyasa's father was a Brahmin sage Parâshara, and mother Matsyagandhâ, the virgin daughter of a fisherman. (5) Kripa s father was a Brahmin sage, Sharadvân Gautama, and mother the goddess Jânapadi. (6) Drona's father was the Brahmin sage, Bharadvâja, and mother the goddess Ghritâchi. (7) Karna's mother was Kunti, who conceived during her maidenhood, and father the god sun. For detailed information vide the accounts of their births: for (1), see chapter 174, Âdiparva, Mahabharata, or in Rigveda, 7, 33, 11-13; for (2), chapter 6, Skandha I, Srimad Bhagavata, for (3) section 4 Prapâthaka iv, Chhândogya Upanishad; for (4), (5), (6) and (7) chapters 105, 130, 130 and 111, respectively of the Âdiparva of the Mahabharata.
*footnote 2: In her anxiety to save her reputation, Kunti threw the newborn child Karna, into water. A charioteer found the child in his pitiable condition and adopted him as his son.
All enlightened people other than Brahmin born are examples of Varna migration. Because only Brahmin is capable to become enlightened. One takes many birth to become Brahmin, some (however few) become Brahmin in single birth.
One must have these qualities to become enlightened.
The actions of a brahmana arising from his own nature are serenity, self-control, austerity, purity, tolerance, honesty, knowledge of the Vedas, wisdom and firm faith. BG 18.42.
Same concept is beautifully expressed in the commentary of Yathartha Geeta.
Ancient sages divided the travellers on the eternal path of the Self into four classes, Shudr, Vaishya, Kshatriy, and Brahmin, according to their innate abilities. In the primary stage of accomplishment every seeker is a Shudr, meaning one who is deficient in knowledge. He spends hours on worship and adoration of God, and yet fails to render even ten minutes of his time truly beneficial to his spiritual quest. He is unable to cut through the illusory facade of the material world. Sitting devoutly at the feet of a realized Sage, an accomplished teacher, at this stage helps in the cultivation of virtues in his nature. With this he is promoted to the level of a seeker of the Vaishya class. Gradually realizing that accomplishments of the Self are the only true accomplishments, he becomes adept in seizing and protecting his senses. Passion and wrath are fatal to the senses, whereas discrimination and renunciation protect them, but they are by themselves incapable of annihilating seeds of the material world. Gradually, then, as the worshipper progresses further, his heart and mind grow strong enough to carve their way through the three properties of nature. This is the inborn quality of a Kshatriy. At this point the worshipper acquires the ability to destroy the world of nature and its perversions. So this is the point of commencement of the war. By further refinement after this, the worshipper is slowly elevated to the category of a Brahmin. Some of the virtues that now grow in the seeker are control of the mind and senses, incessant contemplation, simplicity, perception, and knowledge. By slowly perfecting these qualities, then, he ultimately attains to God, and at this stage he ceases to be even a Brahmin.
Krishn lays down that even if the inherent ability, by which a man participates in this dharm, is of the unmeritorious and ignorant Shudr level, it is beneficial in the highest sense, because it is the starting point from where he can set out on the path of Self-cultivation. However, the worshipper is destroyed if he imitates the manner of higher classes.
Conclusion- Order of progress to become one with ultimate reality is: Shudra -> Vaishya -> Kshatriya -> Brahmin -> Beyond 3 Gunas.
Many take many births to go beyond 3 Gunas from Shudra, some can go beyond 3 Gunas from Sudra in single life as well via dedicated intense Spirituality & devotion.
No, it is not.
According to Krishna in the Gita, one's varNa is a consequence of his innate guNas (pre-natal qualities attached to the soul or AtmA), which are determined by his actions in past births (karmas), over millions of past births. The guNas that a person possesses at the time of his death determine his varNa in the next birth - if he is dominated by sattva (goodness), then he is born as a Brahmana; if he is dominated by rajas (passion, activity), then he is reborn as a Kshatriya; if he has a mixture of rajas and tamas (darkness, ignorance, laziness, superstition), but with rajas dominating, he is reborn as a Vaishya; and if he has a dominance of tamas with some rajas, then he is born as a Shudra. If his soul is completely enveloped in tamas, he will be reborn as an animal.
This is done so that in his next birth, his AtmA is born into an environment which matches the guNas he possesses. According to Krishna's teaching, Brahmanas possess sattva in abundance, and therefore a soul that dies with sattva as dominant will be born into a family of Brahmanas. Kshatriyas possess rajas in abundance, and so an AtmA dying in the mode of rajas is reborn as a Kshatriya, to provide a proper match for its guNas; and so on.
A change in varNa is only possible on rebirth. One cannot short-circuit this process. This is illustrated very well by the story of Matanga, mentioned in the Mahabharata. Matanga was born a Chandala and abandoned as a baby. A childless Rishi adopted the abandoned newborn and raised him as his own child. As a teenager, he learned the truth of his low birth, and resolved that he would correct it. He performed intense penance again and again and invoked Indra, the king of the Gods, and asked him for a boon that he would be converted to a Brahmana. Indra said this was absolutely impossible, and told him he would have to go through trillions and quadrillions of births in other varNas before he could become a Brahmana, even though Matanga was by now a purified soul because of all the penance he had done, and had achieved complete mastery over his senses.
Read about the story here: http://www.sacred-texts.com/hin/m13/m13a027.htm
The other examples people will give you, such as Parashurama, Vishwamitra, etc., are all special cases, and do not represent the standard rule. You can read about those here: http://www.leftbrainwave.com/2017/01/the-scriptural-sanction-for-caste-based.html#TOC5c2f
I hope that answers your question.