The coming of Lord Vishnu's 10th and final avatar, Kalki, has been long foretold (see this answer for detail).
I am curious as to whether any other avatars were similarly foretold in advance. If so, where did the prophecies come from, and how did people at the time react to them?
The coming of different Avatars has been foretold in different ways. For example, in the Valmiki Ramayan when Maharishi Gautam cursed his wife Ahalaya, he also told her that she would be relieved of the curse only after the coming of Dashrath's son Rama to the hermitage:
tathaa shaptvaa ca vai shakram bhaaryaam api ca shaptavaan | iha varSa sahasraaNi bahuuni nivaSisyasi || 1-48-29 vaayu bhakSaa niraahaaraa tapyantii bhasma shaayinii | adR^ishyaa sarva bhuutaanaam aashrame asmin vaSisyasi || 1-48-30
"On cursing Indra thus the sage cursed even his wife saying, 'you shall tarry here for many thousands of years to come without food and consuming air alone, and unseen by all beings you shall live on in this hermitage while contritely recumbent in dust. [1-48-29, 30]
yadaa tu etat vanam ghoram raamo dasharatha aatmajaH | aagamiSyati durdharSaH tadaa puutaa bhaviSyasi || 1-48-31
'When that unassailable son of Dasharatha, namely Rama, arrives at this squalid forest, for it will be henceforth rendered so along with you, then you will be purified. [1-48-31]
Similarly, according to Shrimad Bhagvat Puran when Narad Muni cursed the twins Nalkuber and Manigriva who were turned itno Arjuna trees, he told them that they would be released from the curse after the advent of Shri Krishna:
King Parikshit asked Shukdev about the trees. Shukdev said: "Parikshit, the lord of wealth Kuber had two sons Nalkubar and Manigreev. One day they were enjoying the sweet company of pretty women on the bank of Mandakini river. Just by coincidence, Devarshi Narad arrived there. Out of Shyness, the women folk at once covered themselves, but both the sons of Kuber stood boldly without feeling any shame. Indignant Narad cursed them to become trees and stay in that form for one hundred years. Narad showed kindness as well that despite being in tree forms, they would have the memory of God alive and would be saved by Lord Shri Krishna.
Even in the story of the Suryavanshi king Nriga he was told by Yamraj that he would return to his original state after being rescued by Lord Krishna after a thousand years. As mentioned in the Mahabharat:
Some time before, certain young men of Yadu's race, while searching for water, had come upon a large well covered with grass and creepers. Desirous of drawing water from it, they laboured very much for removing the creepers that covered its mouth. After the mouth had been cleaned, they beheld within the well a very large lizard residing within it. The young men made strong and repeated efforts for rescuing the lizard from that situation. Resembling a very hill in size, the lizard was sought to be freed by means of cords and leathern tongs.
Vasudeva then proceeded to the spot and took out the lizard and questioned it as to who it was. The lizard said that it was identical with the soul of king Nriga who had flourished in days of old and who had performed many sacrifices.
Unto the lizard that said those words, Madhava spoke, 'Thou didst perform many righteous acts. No sin didst thou commit. Why, then, O king, hast thou come to such a distressful end?
To cut the long story short, the king had donated uncountable numbers of cows to the Brahmins after a sacrifice but once one of them managed to re-enter his cowshed cowshed and was gifted to a different Brahmin. Both the Brahmins had an argument and went away unsatisfied. After his death Nriga was adjudged by Yamaraj thus:
Worshipping me duly Yama addressed me, saying, 'The end cannot be ascertained, O king, of thy deed. There is, however, a little sin which was unconsciously perpetrated by thee...... I answered, saying, 'I shall first undergo the distress of punishment, and when that is over, I shall enjoy the happiness that is in store for me, O lord!'
After I had said those words unto the king of the dead, I fell down on the Earth. Though fallen down I still could hear the words that Yama said unto me very loudly. Those words were, Janardana the son of Vasudeva, will rescue thee upon the completion of a full thousand years, when the demerit will be exhausted of thy sinful act, thou shalt then attain to many regions of inexhaustible felicity that have been acquired by thee through thy own acts of righteousness.
Thus you can see that besides Kalki, there have also been prophecies of appearance of other incarnations of Lord Vishnu.
Yes, Sri Caitanya Mahaprabhu.
Predictions of the appearance of Lord Caitanya can be found in many Vedic texts. One of the oldest prophecies concerning Sri Caitanya’s appearance in this world is found in the Atharva-veda verse, starting as:
Ito ‘ham krita-sannyaso ‘vatarisyami.
Meaning:
“I will descend as a sannyasi, a tall, fair, and saintly brahmana devotee, after four to five thousand years of Kali-yuga have passed. I will appear on earth near the Ganges shore and with all the signs of an exalted person, free from material desires. I will always chant the holy names of the Lord, and, thus, taste the sweetness of My own devotional service. Only other advanced devotees will understand Me.”
Also, in a verse from the Sama-veda, starting as:
Tathaham krita-sannyaso bhu-girbanah avatarisye,
Meaning:
The Supreme Being says that He will descend to earth as a brahmana-sannyasi at a place on the shore of the Ganges. Again and again He will chant the names of the Lord in the company of His associates to rescue the people who are devoured by sins in the age of Kali.
The Mundaka Upanishad (3.3) also relates the prophecy of Sri Caitanya in a different way. It states,
“When one realizes the golden form of Lord Gauranga, who is the ultimate actor and the source of the Supreme Brahman, he attains the highest knowledge. He transcends both pious and impious activities, becomes free from worldly bondage, and enters the divine abode of the Lord.”
Another prophecy of the appearance of Sri Caitanya is found in two verses in the Bhavishya Purana. It states:
ajayadhvamaja yadhvam na sansayah kalau sankirtana rambhe bhavisyami saci sutah
Meaning:
"The Supreme Lord said: 'In Kali-yuga, I will appear as the son of Saci, and inaugurate the sankirtana movement. There is no doubt about this.'"
anandasru-kala-roma-harsa-purnam tapo-dhana sarve mam eva draksyanti kalau sannyasa-rupinam
Meaning:
"O sage whose wealth is austerity, in the Kali-yuga everyone will see My form as a sannyasi, a form filled with tears of bliss and bodily hairs standing erect in ecstasy."
Another is from the Svetasvatara Upanishad (3.12):
mahan praburvai purushah sattvasyaisha pravartakah sunirmalamimam praptim ishano jyotiravyayaha
Meaning:
'He is the most Benevolent Supreme Divinity [Mahaprabhu or the great master], as [through sankirtana] He graciously instigates [or bestows] intuitive wisdom in the jiva soul unto its fully developed cognition or purest attainment. This attainment of purest state or immortality is possible only by His grace as He is the Supreme Propeller and Imperishable Transcendental Enlightening Force.'
Another is from the Vayu Purana:
"In the age of Kali I shall descend as the son of Sachidevi to inaugurate the sankirtana movement."
This is also confirmed in the Srimad-Bhagavatam (11.5.32) where it states:
“In the age of Kali, intelligent persons perform congregational chanting to worship the incarnation of Godhead who constantly sings the names of Krishna. Although His complexion is not blackish [like that of Lord Krishna], He is Krishna Himself. He is accompanied by His associates, servants, weapons and confidential companions."
Source: http://www.stephen-knapp.com/sri_caitanya_mahaprabhu.htm
