We know that at the end of every Dwapara yuga, Vyasa compiles the events of the yugas into various puranas. These puranas sometimes contains stories and other details about Kritayuga, Dwapara Yuga and Treta Yuga of different Mahayugas also. But I wonder why I have not come across any stories from any previous Kaliyuga. At the most, what we read about is the stories around starting of Kaliyuga like those related to Parikshit or Janamejaya, etc but not later part of Kaliyuga. Do we know why is it so? Is it that there are no incarnations in later part of any previous Kaliyuga, therefore there were no stories, or are there any stories which I am unaware of?
There are multiple incarnations of Vishnu which took place during previous Kali Yugas:
- Pramati: Just as Vishnu'e incarnation Kalki will come at the end of the present Kali Yuga and kill all the evil people, during one of the Kali Yugas of the Swayambhuva Manvantara Vishnu incarnated as a king named Pramati and killed all the evil people, as described in this excerpt from the Matsya Purana:
King Pramati of the Bhrigu family and Chandramasa gotra was born at the end of Swayambhuva Manu in the Sandhyamsha period, to inflict proper punishment on the sinners. That king travelled all over the earth for 30 years and collected arms and ammunitions, elephants, horses and chariots and marched with a vast army consisting of horses and elephants and accompanied by 100,000 Brahmana soldiers armed with various kinds of weapons, against the Mlecchas and destroyed them. After killing all the Sudra kings he annihilated all the hypocrites After destroying all the sinners and subduing the people living in the North, central regions, the mountaineers, the inhabitants in the East and West, the residents on the Vindhyas, the Deccanis, the Dravidians, the Singhalese, the inhabitants of the Mleccha countnes...
King Pramati was born of Vishnu's part in Manu's family and was famous as Chandramasa. He roamed about the Earth for 20 years and killed all the wicked men in his 32nd year.
- Parashurama: Vishnu's incarnation Parashurama killed the kings of the earth 21 times, thereby bringing about the end of the Kali Yuga and beginning of the Satya Yuga, as described in this chapter of the Drona Parva of the Mahabharata:
Even the great ascetic Rama, the hero worshipped by all heroes, that son of Jamadagni, of great fame, will die, without being contented (with the period of his life). Rooting out all evils from the earth, he caused the primeval Yuga to set in. Having obtained unrivalled prosperity, no fault could be seen in him. His father having been slain and his calf having been stolen by the Kshatriyas, he without any boast, slew Kartavirya who had never been vanquished before by foes. With his bow he slew four and sixty times ten thousand Kshatriyas already within the jaws of death. In that slaughter were included fourteen thousand Brahmana-hating Kshatriyas of the Dantakura country, all of whom he slew. Of the Haihayas, he slew a thousand with his short club, a thousand with his sword, and a thousand by hanging. Heroic warriors, with their cars, steeds, and elephants, lay dead on the field, slain by the wise son of Jamadagni, enraged at the slaughter of his father. And Rama, on that occasion, slew ten thousand Kshatriyas with his axe.
By the way, see my answer here for what happened after Parashurama killed all those kings.
- Matsya: In the last Kali Yuga of the Chakshusha Manvantara, Vishnu's incarnation Matsya the fish rescued the king Satyavrata, also known as Vaivasvata Manu, from a flood that comes at the end of every Manavantara, as described in this chapter of the Srimad Bhagavatam:
When King Satyavrata spoke in this way, the Supreme Personality of Godhead, who at the end of the yuga had assumed the form of a fish to benefit His devotee and enjoy His pastimes in the water of inundation, responded as follows. The Supreme Personality of Godhead said: "O King, who can subdue your enemies, on the seventh day from today the three worlds — Bhūḥ, Bhuvaḥ and Svaḥ — will all merge into the water of inundation. When all the three worlds merge into the water, a large boat sent by Me will appear before you."
Expect this, this chapter of Skanda Purana mentiones about King Ama, who lived at one of previous Dwapara-Kali yuga transtition.
12-13. Now, when the advent of Kali Age was imminent, at the end of Dvāpara, when Kali had not yet started, there was a king named Āma who became the ruler of Kānyakubja. He was glorious, a knower of Dharma, calm, self-controlled, eager to maintain justice, of good conduct and devoted to truth and piety, O excellent king.
To answer your other question, the reason why our scriptures only discuss the beginning of the present Kali Yuga is that that's when the Puranas were composed. So any description of events after that are only in the form of prophecies, for instance the Vishnu Purana's prophecy of the Janamajeaya's descendants discussed in my answer here.
As far as I know, the chronologically latest event described (in the past tense) in Hindu scripture, excluding Tamil scripture like the Alwars' poems, is a Yagna conducted during the reign of Janamejaya's great-grandson Adisimakrishna, which is described in this excerpt from the Vayu Purana. In fact, it is during this Yagna that the Vayu Purana was recited by Ugrashrava (aka Suta).
Here is one event that happened in a previous kaliyuga. From Skanda Purana, Pursotama-kshetra-Mahatmy, Section 2
21-23. In a former Kaliyuga, O Brāhmaṇas, the Prajāpati (Patriarch) named Dakṣa was moved with pity on seeing men suffering from distresses coming from bodies and other sources. He went there and celebrated the great festival in the manner described by me. He was the first person to do so. On the third day in the bright half of the month of Vaiśākha he applied sandal-paste on the body of the Lord and joyously repeated this prayer:
This section talks about how to perform worship of Lord Pursottama on Akshaya Tritiya. Prajapati Daksha was the first person to do so, and he did it in a previous Kaliyuga