The situation with the Czechs is quite amusing. A small Mongol detachment ventured into the Kingdom of Bohemia and was decisively defeated by superior forces. Since this was the only victory of Europeans over the Mongols, it was greatly exaggerated. In reality, the Mongols were simply pursuing a different objective – to finish off the Cumans who had fled to Hungary. And, incidentally, others suffered as a result.
As for the overthrow of the Mongol yoke, that also turned out to be quite peculiar. After the Golden Horde disintegrated into several warring states (the Kazan Khanate, the Astrakhan Khanate, the Crimean Khanate, etc.), there was no longer a single enemy. The yoke simply disappeared on its own, without any intervention from the Russians (the Stand on the Ugra River doesn't count, as there wasn't even a battle). The fragments of the Mongol states gradually became part of Russia, without even putting up much resistance. They simply perceived the Russian Tsar as the legitimate heir of Genghis Khan (which, in principle, was the case, considering that they were so closely related that 60-80% of the Russian aristocracy had Mongol-Tatar roots). The last remnants of the Mongol khanates became part of Russia only in the 19th century (the Kazakhs).