The Kara-Kitai were one of the steppe people in Central Asia.
1234, the Kara-Kitai were defeated by the Mongols. Ogadai now reigned the biggest (if sparely populated) empire on the planet. But as soon as 1241, Ogadai died, and the designated successor, his son Guyuk couldn't hold them together. Jalal-ad-Din could use the situation for a counter-attack and drove the Mongols back behind Lake Balchash. The steppe people once again fell apart, not to be united at least for decades, waiting for another strong leader.
During the 1240s however, some Kara-Kitai tribes displaced by the Mongols defeated the eastern Kipchaks, crossed the lower Volga, attacked the area south of Don and Volga and destroyed the little country of the Alans, a leftover of the völkerwanderung. In 1246, Baibars, who became a Kipchak leader, defeated the Kara-Kitai, threw them back behind the Don.
After Shah Jalal-ad-Din of Choresm died in 1255, some areas of his empire (Kara-Kitai, Afghanistan) tried to break away. The Kara-Kitai managed to stay independent, while Afghanistan was pacified in the following years and stayed in the fold.
When in the 1380s steppe people leader Arik-Buqa drove the Kara-Kitai and other Siberian people west, they also harassed the Volga area. That's why the eastern Russian princes made Vsevolod V Yuryevich of Vladimir-Suzdal their leader, to fight them.