The core population likely didn't move that far initially. The evidence we have suggests that Cahokia's population was siphoned off to increasingly more distant outlying sites over the course of about three generations. As Cahokia's power wanes, the Angel Site in southern Indiana becomes the dominant force in the Middle Mississippian world (though it's never as prominent as Cahokia was). Around 1450, the people of Angel close up shop rather abruptly and establish a collection of towns around the mouth of hte Wabash, probably forming a confederacy.
Both Cahokia and Angel, along with many other Middle Mississippian sites are believed to be ancestral, by and large, to the Dhegiha. Through various mergers and divergences, this gives rise to the five historic and current Dhegiha nations: the Osage, the Omaha, the Kansa, the Ponca, and the Quapaw.
There's also a strong possibility that Cahokia was a multi-ethnic, multi-lingual community. Thomas Emerson, one of the leading archaeologists studying Cahokia, argues that up to a third of Cahokia's population was immigrants from outside the American Bottoms and, in Beyond Collapse, that the Cahokia dissolved due to an inability to maintain that large diverse population. Personally I'm skeptical of any silver bullet to explain the dissolution of Cahokia, but Emerson is almost certainly right about it being a diverse place. This was true of most large Native communities in the historic era too. So there were likely people from all over who either went back to live with more closely related peoples or became assimilated with the locals. The opposite is, of course, true as well. Some Dhegiha peoples from Cahokia were likely absorbed into the expanding Illinois Confederacy in more recent history. So while the Algonquian-speaking Cahokia who lived in the area historically and from whom the site was eventually named don't have a particularly strong culture connection to the site, they could well have had ancestors who lived there, one way or another.
In addition to the Emerson article / chapter mentioned earlier, I'd recommend checking out Cahokia: The Process and Principles of the Creation of an Early Mississippian City in Making Ancient Cities: Space and Place in Early Urban Societies, Cahokia Interaction and Ethnogenesis in the Northern Midcontinent (also by Emerson) in The Oxford Handbook of American Archaeology, and really anything by Timothy Pauketat (Cahokia: Ancient America's Great City on the Mississippi is the most accessible to a general audience, but Ancient Cahokia and the Mississippians, while a bit more technical, is a quick easy read as well if you want a crash course in Cahokia - though it could stand a new edition).
Question since I'm ignorant, why exactly was Cahokia one of the most impressive cities? Presumably there was something special about it?
Excavated there as a volunteer, I only mention this in case others want to do so as well. They offer sessions for anyone that wants to help. You can even go for only a few days if you want. I will warn though some of the summers were brutal. =P
Is there any evidence that Cahokia was in fact a city, and not something more similar to a ceremonial meeting place of a conglomerate culture?
Pardon my ignorance, I have never heard of Cahokia or frankly any medieval history in the modern day United States. If I may, as a follow-up question, ask, why did I, as a born and raised Midwesterner, never learn of any medieval history of the U.S. much less have ever even had such a thing suggested?