Hi. I've researched the period in question however will only be able to give a relatively vague answer.

If you'd like to know more about this subject I can't recommend reading Donald Ostrowski more - he is probably the leading historian in the field concerning the end of Kievan Rus and the Mongol Invasion. Also Robert Crummey is very good on this period, as well as Charles Halperin who is also a very credible historian.

Ok - so to the question.

When looking at cultural influences on Muscovy during the 13th century (as well as places like Tver, Novgorod, Riazan, Pereaslavyl and Chernivgov) historians tend to focus on the cultural influence on the Byzantine Empire.

Having already adopted the orthodox religion, Russian princes had far more cultural connections to the Byzantine empire than the did with their Mongol conquerors. This is not to say that the Mongols had no influence, the Yam system was the basis of the Russian postal system. They also had a tremendous influence on the administrative systems in Russia such as taxation systems and they also made censuses of the Russian population.

In many sense they're greatest influence was on the administrative and political systems of Russia, rather than cultural institutions. Mongol art or philosophy did have a great influence on the people, nor did their religion. Nevertheless aspects of the Mongol language did creep into Russian - most notably the word "деньги" (den-gii) which means money.

The reasons why the Mongols didn't have as far reaching cultural influence over Russia as say the Normans had over England is because the Mongols did not invade these towns and assimilate with the Russian people - they largely stayed separate from the Russians in the North (Novgorod, Muscovy, Tver etc.). The Mongols tended to live in what we now call the 'Caucasus' and would simply exact tribute from the Russian populations, invading them on horseback if they refused to pay.

I hope this has provided some insight into Mongol/Russian relations in the 13th century. I'm not an expert on this period, having mainly studied post-Mongol Russia, nevertheless I've read a little into the period and these are some of the key points that I can still remember.