Not sure about Mongol Invasions exactly or if this even answers your question at all, but the Byzantines did employ Turkish & Pecheneg (amongst others of course) mercenaries against the Seljuk dynasty, itself of 'similar background' to them.

Completely offpoint but if one is to believe the records of the battle of Manzikert, these Turkish mercenaries even lasted more at said battle, which was fought against the Seljuk turks, than their Armenian counterparts, which fled.

Yes. The Cumans fled to the kingdom of Hungary after their initial defeat by the Mongols in 1239. King Bela IV agreed to grant them asylum on the condition that the people converted to Christianity and allied with the Hungarians against the Mongols.

Unfortunately the alliance never really came to fruition. When the Hungarian people saw press-ganged Cumans in the invading Mongol army, they got rather concerned they were being tricked. This resulted in a mob killing the the leader of the Cumans, Koten. This resulted in the Cumans departing and I am not sure how many Cumans ended up fighting in the Battle of Mohi.