Hessl’s an idiot about history. Consistently wet weather might have produced rain for the Mongol horseherds, but it would have played hob with the Mongols’ principal weapon - the recurve bow. Mongol bows were made of wood, horn and other materials, held together by a glue made of fish . The glue was not waterproof, and neither were the strings for the bows.
And the Mongol capital wan’t moved to Beiging because of weather. Qublai Quan moved it there after defeating his brother Ariq Boka for the Supreme Quanship of the Mongol empire because: [1] Northern China was part of the Ulus granted him by his brother Mongke Qa Quan, [2] He was also the Emperor of China [at least the Chin part; and he was well on his way to conquering the Sung].
Your point about the glue of the recurve bow being weakened by wet weather is interesting. If so, that would have tended to limit Mongol conquests to areas with arid climates and would account in part for the Mongols’ withdrawal from their conquest of Hungary.