My dear King Parīkṣit, in the next manvantara the lotus-eyed Personality of Godhead Lord Paraśurāma, the son of Jamadagni, will be a great propounder of Vedic knowledge. In other words, he will be one of the seven sages. SB 9.16.25
Lord Paraśurāma still lives as an intelligent brāhmaṇa in the
mountainous country known as Mahendra. Completely satisfied, having
given up all the weapons of a kṣatriya, he is always worshiped, adored
and offered prayers for his exalted character and activities by such
celestial beings as the Siddhas, Cāraṇas and Gandharvas. SB
The Kalki Purana writes that he will reappear at end of time to be the martial guru of Kalki. He will then instruct the kalki avatar to undertake penance in order to receive celestial weaponry, required to save dharma at the end of kaliyuga.
Since your question is more focused on Lord Hanuman let me mention about him first - Shrimad Bhagvat Puran Canto 5 Chapter 19 states that he lives in the Kimpurush Varsha listening to Ramayan being sung by the denizens of the place and himself joining them in
आर्ष्टिषेणेन सह गन्दर्वैरनुगीयमानं
परमकल्याणीं भर्तृभगवत्कथां समुपशृणोति स्वयं चेदं गायति ॥
S'rî S'uka said: 'In the land of Kimpurusha the supreme and greatest devotee Hanumân delighting in service at His feet is together with the people always engaged there in worship of the Supreme Lord Râmacandra, the Original Personality who as the elder brother of Lakshmana is so
pleasing to Sîtâ.
Together with Ârshthishena [the leader of Kimpurusha] attentively listening to the stories about his most auspicious master and Lordship being chanted by a company of Gandharvas, he Hanumân himself prays thus.
The reference about Lord Parshurama has already been given so I won't repeat the same. The Kathasaritsagar mentions the story of a man named Lohajangha reaching Lanka in a flood where the Rakshasas take him to Vibhishan. This story is from a time period after Mahabharat since it mentions Mathura as the city of Krishna:
And finding that he was on the other side of the sea, he was astonished, and looked upon the whole thing as a daydream; then he saw there to his terror two horrible Rākṣasas, and those two for their part contemplated him from a distance with feelings of fear. Remembering how they were defeated by Rāma, and seeing that Lohajaṅgha was also a man who had crossed the sea, they were once more alarmed in their hearts. So, after they had deliberated together, one of them went off immediately and told the whole occurrence to King Vibhīṣaṇa.
King Vibhīṣaṇa, too, as he had seen the prowess of Rāma, being terrified at the arrival of a man, said to that Rākṣasa: “Go, my good friend, and tell that man from me, in a friendly manner, that he is to do me the favour of coming to my palace.”
The Rākṣasa said, “I will do so,” and timidly approached Lohajaṅgha, and told him that request of his sovereign’s. Lohajaṅgha for his part accepted that invitation with unruffled calm, and went to Laṅkā with that Rākṣasa as his companion. And when he arrived in Laṅkā he was astonished at beholding numerous splendid edifices of gold, and entering the king’s palace he saw Vibhīṣaṇa.
Kripacharya is also mentioned in the Mahaprasthanika Parva of Mahabharat to have become the guru of Parikshit so he was there at least in the beginning of Kaliyug:
"Having said these words, king Yudhishthira the just, along with his brothers, promptly offered oblations of water unto Vasudeva of great intelligence, as also unto his old maternal uncle and Rama and others. He then duly performed the Sraddhas of all those deceased kinsmen of his. The king, in honour of Hari and naming him repeatedly, fed the Island-born Vyasa, and Narada, and Markandeya possessed of wealth of penances, and Yajnavalkya of Bharadwaja’s race, with many delicious viands. In honour of Krishna, he also gave away many jewels and gems, and robes and clothes, and villages, and horses and cars, and female slaves by hundreds and thousands unto foremost of Brahmanas. Summoning the citizens, Kripa was installed as the preceptor and Parikshit was made over to him as his disciple, O chief of Bharata’s race.
Others are supposed to appear in the future but not sure if they would appear or show some signs of their existence in the Kaliyug itself. For example, Ashwatthama is supposed to become the next Vyas according to the Vishnu Purana:
These are the twenty-eight elder Vyásas, by whom, in the preceding Dwápara ages, the Veda has been divided into four. In the next Dwápara, Drauńi (the son of Drońa) will be the Vyása, when my son, the Muni Krishńa Dwaipáyana, who is the actual Vyása, shall cease to be (in that character).
Similarly, as mentioned in this chapter of the same Purana, he shal also be one of the Saptarishis and Maharaj Bali shall become the next Indra but all that is in the next Manvantar:
The son of Chháyá, who was called also a Manu, was denominated Sávarńi 6, from being of the same caste (Savarńa) as his elder brother, the Manu Vaivaswata. He presides over the ensuing or eighth Manwantara; the particulars of which, and the following, I will now relate. In the period in which Sávarńi shall be the Manu, the classes of the gods will be Sutapas, Amitábhas, and Mukhyas; twenty-one of each. The seven Rishis will be Díptimat, Gálava, Ráma, Kripa, Drauńi; my son Vyása will be the sixth, and the seventh will be Rishyasringa. The Indra will be Bali, the sinless son of Virochana, who through the favour of Vishńu is actually sovereign of part of Pátála. The royal progeny of Sávarńi will be Virajas, Arvarívas, Nirmoha, and others.
As per the biography of Gosvami Tulsidas by A.P.N. Pankaj(Sri Ramakrishna Math, Mylapur)
Sri Hanuman had sought and received from Sri Rama the boon that he may live forever on earth and be present wherever there is 'Raghunatha-sankirtanam'(Ananda-Ramayana I.12.141-45). A preta..got pleased with Tulasi..told that Sri Hanuman attended Tulsidas's discourses in the guise of a leper...Next day,Gosvamiji recognised Sri Hanuman in the audience attending his discourse, caught hold of his feet..and entreated him to help him have Sri Rama-darshana.(Page 27)
Sant Tulsodas(1532-1623) lived very much in the Kali yuga. This proves that Hanumanji appeared in the Kali yuga.Sri Neev Karori Baba (1900-1973, https://en.m.wikipedia.org/wiki/Neem_Karoli_Baba )is also known to have got the darshan of Hanumanji.
Eight immortals still exist in this Kaliyuga and would be playing important roles in coming future.
Ashwatthama and Krishńa Dwaipáyana
Krishńa Dwaipáyana still holds the title of Vyasa and Ashwatthama will be holding the title of Vayasa in 29th Dwapara Yuga:
In the next Dwápara, Drauńi (the son of Drońa) will be the Vyása, when my son, the Muni Krishńa Dwaipáyana, who is the actual Vyása, shall cease to be (in that character). ~ Vishnu Purana: Part 1: Chapter 3
Ashwatthama, Krishna Dvaipāyana, Kripacharya and Parashurama
Ashwatthama and Krishna Dvaipāyana will be included in Saptrishsis of 8th Manvatara along with Kripacharya and Parashuram:
The seven Rishis will be Díptimat, Gálava, Ráma, Kripa, Drauńi; my son Vyása will be the sixth, and the seventh will be Rishyasringa. ~ Vishnu Purana: Part 1: Chapter 2
He will hold the title of Indra from current Indra named Purandra in 8th Manavantara by the grace of Lord Vishnu:
The Indra will be Bali, the sinless son of Virochana, who through the favour of Vishńu is actually sovereign of part of Pátála. ~Vishnu Purana: Part 1: Chapter 2
Hanumana and Vibhishana
They are supposed to live on earth till the history of Lord Rama remains current:
He then said to Vibhishana the king of Rakshasas: As long as people shall breathe, O Vibhishana, O highly powerful lord of Rakshasas, thou shalt preserve thy body in Lanka.
As long as the moon, the sun and the earth shall exist and as long as my story shall remain current, thy kingdom shall be in existence. Thou didst, out of fiendship, carry out my behests and perform my works. Do thou piously govern thy subjects.
I should not speak any thing else. What shall I say more unto thee, O highly powerful lord of Rakshasas. Worship Jagannatha (the lord of the world), the presiding deity of the Ikshwaku race.
Having thus addressed the king of Rakshasas always obeying Raghava s command, Kakuthstha said to Hanuman "It is settled that thou shalt live forever, do thou, now observe thy promise. As long as my history shall run current in this world do thou at my command live happily." Being thus addressed by the high souled Raghava Hanuman attained to great delight and said: As long as the sacred theme shall pass current in this world I shall live here carrying out thy commands.
Thereupon he said to the Jambavan, Brahma s son, Mainda, Dvivida and five others in his company: "As long as the Kali Yuga exists do ye all live." ~Valmiki Ramayana: Uttarakhand
For 3,000 years thou shalt wander over this earth, without a companion and without being able to talk with anyone. Alone and without anybody by thy side, thou shalt wander through diverse countries, O wretch, thou shalt have no place in the midst of men. The stench of pus and blood shall emanate from thee, and inaccessible forests and dreary moors shall be thy abode! Thou shalt wander over the Earth, O thou of sinful soul, with the weight of all diseases on thee.
So Ashwathama was born immortal and is wandering around the earth because of Shri Krishna's curse.
When Rama was about to leave Ayodhya at the end of his reign, Lord Rama in his original form of Shri Vishnu ordered Vibhishana to stay on earth and serve the people and guide them to the path of truth and Dharma. Hence, Vibhishana is considered one of the seven immortals or Chiranjeevins.
Rama cured him and blessed him with immortality, but Hanuman refused this and asked only for a place at Rama's feet to worship him. Touched, Rama blessed him with immortality anyway. Like Shesha Nag, Hanuman would live on after the Kalpa (destruction of the universe).
Hanuman prophesied that Bhima would soon be a part of a terrible war, and promised Bhima that he would sit on the flag of his brother Arjuna's chariot and shout a battle cry for Bhima that would weaken the hearts of his enemies. Content, Hanuman left his brother to his search, and after that prophesied war, would not be seen again until early 1600s when he met Tulsidas.
He will appear at the end of the Kali yuga to be the guru of Vishnu's tenth and last avatar, Kalki. He will instruct Kalki.
Diptimat, Gslava, Parasurama, Kripa, Drauni or Ashwatthama, Vyasa and Rishyasringa will be the Saptarishis of Savarni manvantara and Kripacharya, Parashurama, Vyasa, and Ashwathama are chiranjivis.
King Bali was pushed by the pious Vama to Sutala Loka. Vishnu, impressed with his devotion blessed him with boon of immortality and told him to rule Patala Loka. He also told him that he will be Indra in Savarni Manvantara. So as of now, he is in Sutala Loka ruling there and will reside in Swarga in his next manvantara when he becomes Indra. Sutala Loka is a place just below Patala Loka and is even more beautiful than Swarga. On Onam,Bali returns to earth to visit his people in Kerala. King Bali is son of Virochana, grandson of King Prahlad, and great-grandson of Hiranyakashipu.
Besides Ashwathama who continues to wander around the earth and Parashurama who resides in Mahendragiri Mountains and King Bali who was sent to Sutala Loka by Vamana, the rest of the chiranjivis that include, Vyasa, Hanuman, Kripacharya, Markandeya, and Vibhishana continue to reside and live in Himalaya Mountains.
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