The most famous is Koten's horde that migrated to Hungary after the defeat at Kalka, supposedly 40,000 "families". Supposedly they all got baptized (Koten himself in 1239). His daughter married the king's son, but the Hungarian nobles grew suspicious of them and killed many in a pogrom in the 1241. The Cumans pillaged the country in return, then most migrated to most then migrated to Bulgaria where they were a key component of future Bulgarian nobility and army.
It's likely that the Asens, Shishmans, Terters and probably Basarab himself were of Cuman ancestry.
Some thousands remained settled in Georgia, some probably relocated to Vladimir Rus (there are some place names that indicate Black Hats, so it's possible that Cumans also settled there), many were killed (resistance continued after '37, last major leader in the chronicles was Bachman who was killed in 38-39 during the invasion of Bulgaria). Most probably got assimilated by the Horde.
There are Kypchak/Cuman tribal names among Kazakhs, (modern) Kyrgyz, Nogai, Bashkirs and of course there's the Kunszag in Hungary.
So they probably remained pretty widespread but not very numerous. Batu notoriously had a hard time gathering a large army from his ulus after the Mongol empire became permanently disunited.