Why did the main battles with the Tatar-Mongols and their movements take place in the winter?

Because at that time the rivers were covered with ice. Thus, the Tatar cavalry could easily cover long distances. The Mongol-Tatar bulk was the reason for the final fall of Russia, which began with internal strife in the 11th century. So the Russian lands were in the dominion of the horde. The khan imposed a large tribute on the Russian principalities, and the princes were forced to recognize vassal dependence.