Well, as I know Bashkirs were originally Finno-Uralic nomads a kin to Hungarians, however while Hungarians had moved to the West, Bashkirs stayed at their original land which is located to the South of Ural mountains.
Bashkirs were powerful warriors and they were able to hold the horde of Mongolian Batu-khan for a number of years in the 13th century A.D. before he finally made it to Russia and Eastern Europe.
After the Mongol conquest, Bashkirs became a part of the Golden Horde and were culturally assimilated by Turkic culture which was dominant in this state. Thus, they gradually adopted Turkic language and customs as well.
I think Hungarians remembered for several centuries about some of their people left in the East. Even several Catholic envoys were sent to the land of Bashkirs with in order to bring them to Hungary, however these missions were not succesful, most likely due to the opposition from the rulers of the Golden Horde.
Although, linguistically and culturally Bashkirs are Turkic now, there are, as I know some tensions with the neighboring Kazan Tatars, which can be attributed to the historical past.In other words, Bashkirs and Tatars do not like each other very much. In 18 and 19 centuries there were also constant military conflicts between Kazakhs Turks and Bashkirs.
Since, Kypchak was the dominant language of the Golden Horde it is natural that modern Bashkir originates from Kypchak. In fact, until the beginning of the 20th century the high society of Bashkirs used the language of Kazan Tatars and only during the Soviet time Bashkir was recognized as a separate language.