We have the most information on the fates of Nova Cats and Wolves, because they formed coherent splinter factions that were a major focus of the Dark Age game and era. We can't really track them by unit, because the front line forces were intentionally mixed together and integrated with the House and Comstar units to form the RAF and their Hastati, Principes, and Triari Brigades. Though we do know that each Clan community did form militias that would serve as parts of the Republic Standing Guard, because Kal Radick got his start commanding a Clan Wolf militia unit known as the Bright Fang. It was in fact the Clan militias of Prefecture IV that would form the core of the Steel Wolves.
Many of the Nova Cats formed the Spirit Cats that would eventually settle in the Free Worlds League and help form the Clan Protectorate, some of them also traveled to the Nova Cats of the Draconis Combine to convince them to join them, though only Kisho Nova Cat's remnant was able to escape the Nova Cat annihilation after they supported Emi Kurita and Katana Tormark over Yori Kurita. Also, many Nova Cats remained with the Republic up until its end including a Paladin who died in the fighting for Terra.
The Wolves formed the Steel Wolves around Kal Radick, who was eventually killed by Anastasia Kerensky, who took them for her own. She eventually would form a mercenary force known as the Wolf Hunters, which would shed many Steel Wolves. By the ilClan era many, if not most, of the Steel Wolves have made their way to Clan Wolf proper, with many of those that Anastasia Kerensky had seeded throughout the Inner Sphere now being recalled to serve her and Clan Wolf, if the short story Wars and Rumors is to be believed. Like the Nova Cats many Wolves remained loyal to the Republic, including a freeborn of Irish/Wolf Elemental stock, known as Padraig, who lived on Skye.
The Ghost Bears we don't hear much of, though it could be likely they helped to form the Vega Protectorate, and were thus absorbed into the Rasalhague Dominion.
The Falcon and Horse descendants we know the least about, especially since they seem to have been the least numerous of the Clan settlers in the Republic.