Advaita tells no such thing - it is "Advaita Vedanta" AKA SMARTISM or smriti following sect that tells varna is based on birth because this is the command of semi-sacred Smritis. All the sacred Hindu scriptures fully support change of Varna. Vaishnavas (except Ramanuja) like Alvars, Gaudiya & Ramanandi are fully in agreement that Varna can be changed because it is based on Vaishnav Purans. There are over 100 examples of Varna migration in Srimad Bhagvat Puran itself:
SB 7.11.35 - If one shows the symptoms of being a brāhmaṇa, kṣatriya, vaiśya or śūdra, as described above, even if he has appeared in a different class, he should be accepted according to those symptoms of classification.
SB 7.11 calls 2nd birth into higher Varna as purification... and says in 2.4 that all kinds of people like "Kirāta, Hūṇa, Āndhra, Pulinda, Pulkaśa, Ābhīra, Śumbha, Yavana, members of the Khasa races and even others addicted to sinful acts can all be purified"
SB 9.2 - From the kṣatriya known as Agniveśya came the celebrated brāhmaṇa dynasty known as Āgniveśyāyana
SB 5.4.13 - There were EIGHTY-ONE younger ones, all born of Ṛṣabhadeva and Jayantī..... They all became perfectly qualified brāhmaṇas [from kṣatriyas].
SB 9.2.17 - From the son of Manu named Dhṛṣṭa came a kṣatriya caste called Dhārṣṭa. Although the Dhārṣṭas belonged to the kṣatriya caste, they were able to convert themselves into brāhmaṇas.
There are too many examples to count !! Now moving on to Vishnu Puran it also has too many examples. Here are a few:
Prishadh was son of King (Kshatriya) Daksha but became a Shudra. Further he did Tapasya to achieve salvation after repenting. (Vishnu Puran 4.1.14, Parasara)
The sons of Nabhag became BOTH Brahmin and Kshatriya (Vishnu Puran 4.2 beginning)
Various different kshatriyas became mlecchas (Vishnu Puran, 4.3.18)
Satyavrata was degraded to chandala (Vishnu Puran 4.3.7)
Shaunak became Brahmin though born in Kshatriya family. (Vishnu Puran 4.8.1)
There are too many examples even in other Purans like Brahma Puran. There is no caste in Tantras, and Shaiva Agamas permit equality of caste. There are also at least 7 examples in the Vedas. Rig Veda 9.63 says:
इन्द्रं वर्धन्तो अप्तुर: कृण्वन्तो विश्वमार्यम् । अपघ्नन्तो अराव्णः ॥
This is a prayer to make all humans Aryas. So when Veda is making a distinction between Shudra & Arya it means bad people and noble people. Of course, birth is as per karma but this is a very clear message from the Veda itself that Varna can be changed! Now if we see Upanishads or Brahmanas we almost always get the same picture. Look at Chandogya Upanishad talks about the initiation story of another Upanishad's author:
Gautama asked him, ‘O Somya, what is your lineage?’ Satyakāma said: ‘Sir, I do not know what my lineage is. When I asked my mother, she said to me: “I was very busy serving many people when I was young, and I had you. As this was the situation, I know nothing about your lineage. My name is Jabālā, and your name is Satyakāma.” So, sir, I am Satyakāma Jābāla’. Gautama said to him: ‘No non-brāhmin could speak like this.
So the son of a highly impure shudra woman and unknown man was declared Brahmin!!! Now look at the story of Vedic author Vatsa found in Tandya Brahmana 14.6.6, Jaiminiya Brahmana 3.234:
The two sons of Kanva, Medhatithi and Trioka, contended about the sacred lore (brahman). They said : ' Come, let us cross the flaming fire'. They crossed the flaming fire. Tris'oka crossed over it (unharmed), but of the other it scorched the eye-lashes. Ho (Trisoka) said to him (to Medhatithi) : ' I have vanquished thee '. ' No ', said he, 'thou art the son of an Asura-mother ; even the deities have not wished to touch thee'. Then, they (said): 'Come, let us cross the water'. They crossed the flowing Rathaspa. Tris'oka crossed over it (unharmed), but of the other the rims of his cart were moistened by the water of the Rathaspa (translation uncertain, text corrupted !). He (TriSoka) said to him (to Medhatithi): 'I have vanquished thee'.
let us walk according to the rite through fire (to decide) which of us two is the better brahmin
GOD AGNI preferred Shudra's son. Shudra's son was a superior Brahmin compared to born-Brahmin!!! Another example from the Brahmanas is in Kausitaki Brāhmana 12.3 and Aitareya Brāhmana 8.19
"You are the son of a female slave. We will not eat with you" (dāsya vai tvam putro si na vayam tvaya saha bhaksayisyama iti). Kavasa became angry and ran away. He recited a hymn of praise to Sarasvati, who then followed after him: sarasvatim etena suktena tustava tam heyam iyaya. Realizing what had happened, the surprised seers hurried to him to express their reverence: "0 seer, homage be to you. Do not harm us. You indeed are the best of us...
GODDESS SARASWATI considers a Shudra's son superior to all birth-Brahmins combined at being a Brahmin. The birth-Brahmins also accept they are inferior to him. He authored a part of Rig Veda. There are 108 canonical Upanishads as per Muktika Upanishad. One of them is Vajrasuchika Upanishad which says
It is said that a Brahmana is so because of his caste. This is not acceptable because there are diverse communities in the world... ...Among these many have attained the highest rank, despite of their lower birth and given proof of their wisdom. Therefore a Brahmana is not so because of his community.
Self-explanatory. I have given examples from 7 Vedic texts against caste.
There are too many examples in Itihasas also, but here is the commandment of GOD SHIVA:
Mahabharat 13 SECTION CXLIII - When a pious nature and pious deeds are noticeable in even a Sudra, he should, according to my opinion, be held superior to a person of the three regenerate classes. Neither birth, nor the purificatory rites, nor learning, nor offspring, can be regarded as grounds for conferring upon one the regenerate status. Verily, conduct is the ONLY ground. All Brahmanas in this world are Brahmanas in consequence of conduct. A Sudra, if he is established on good conduct, is regarded as possessed of the status of a Brahmana. The status of Brahma, O auspicious lady, is equal wherever it exists. Even this is my opinion. He, indeed, is a Brahmana in whom the status of Brahma exists,--that condition which is bereft of attributes and which has no stain attached to it. 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.
Shaivism & Shaktism is totally against caste like I have said in Tantras & Agamas. Even examples are in Shaiva scriptures like Puranas -
Shiva Purana's Vidyeśvara-saṃhitā (Chapter 17) is about how Kshatriyas, Vaishyas and Shudras can ALL BECOME BRAHMINS using Shiva's mantra - क्षत्रियः पञ्चलक्षण क्षत्रत्वमपनेष्यति । पुनश्च पञ्चलक्षण क्षत्रियो ब्राह्मणो भवेत् ।। मन्त्रसिद्धिर्जपाद्यैव क्रमान्मुक्तो भवेन्नर । वैश्यस्तु पञ्चलक्षण वैश्यत्वमपनेष्यति ।। पुनश्च पञ्चलक्षण मन्त्रक्षत्रिय उच्यते । पुनश्च पञ्चलक्षण क्षत्रत्वमपनेष्यति ।। पुनश्च पञ्चलक्षण मन्त्रब्राह्मण उच्यते । शूद्रश्चैव नमोऽन्तेन पञ्चविंशतिलक्षत ।। मन्त्रविप्रत्वमापद्य पश्चाच्छुद्रो भवेद्द्विज । नारीवाथ नरो वाथ ब्राह्मणो वान्य एव वा ।।
Shiva Rahasya (secret) Purana 11.109 - As you account a child born into a Caste to belong to that Caste even though he cannot yet perform all that is required of him according to his station, even thus a servant shall be accounted to belong to the Caste whose work he does even though he may not be a full-blood Farmer, Merchant and so forth. A man who, though born into the Priestly Caste, has no knowledge of the Scriptures and does a Farmer's work, cannot be accounted a Priest but only a Farmer. Verily, Caste is determined by the kind of work a man does. And a servant is a man who does another's work; he has no work of his own. Therefore he belongs to that Caste the work of which he does and not to any other. This shall be a settled matter. He who says otherwise sins against My Law.
So actually anyone who insists on caste is actually committing a sin according to Shaivism. EDIT - BIRTH HAS NO PLACE AT ALL IN DETERMINING VARNA! IT IS ONLY CONDUCT WHICH MATTERS And everyone is mixed DNA / mixed up blood. There is no pure gotra/surname even from father's side. This is scientific fact. Advaita Vedanta smarta sect will only listen to smritis and evade other scriptures.
So it is up to people if they want to listen to SHIVA, AGNI, SARASWATI, VISHNU, Vedas, Tantras, Agamas, Itihasas, Puranas etc..... or to Manusmriti !!! And I am not denying birth in a particular community is based on past-karma either !!!
But anyone who says Varna cannot be changed is against Hindu gods and most sacred Hindu scriptures.