Drinking kumis to name just one was brought to Russia by the Mongols. Even in the early twentieth century we have people like Stravinsky going on holiday to southern Russia partially to have kumis as a medicinal treatment, for some sort of lung complaint if I recall correctly. Other examples include the Persian scarves beloved of the Russian aristocracy, many words and phrases, kazna (treasury) is a tartar word for instance, many prominent people claimed ancestry from Mongol forebears, Rimsky-Korsakov, the Godunov family, Bulgakov etc, even the Russian crown the Cap of Monomakh is of Central Asian origin.

Natasha's dance, Orlando Figes

Stravinsky, a life, Stephen Walsh

Mussorgsky and his circle, Stephen Walsh.

Not to cheat him out of an answer, but /u/rakony and myself discussed a similar question about a year ago. If you have further specific questions not answered there or further questions about something discussed there, I'm sure he will be happy, as well as myself, to answer them to the best of our abilities.