I've noticed that the Mongols do not invade other Tengri nations. In my current saved game it was actually game-breaking, as I've played with the Magyars, raided the hell out of my Christian neighbors to the point where I've financed the building of cities in my demesne as well as Centers of trade, managed to reform Tengri over time and installed my own dynasty in most of those areas (giving them independence). Eventually I was so rich that once the religious head called for a Great holy war against Kievan Rus, I conquered it, installed my second son as the ruler, then waged holy wars within the Rhos de-jure territory, and eventually gave independence and money so they can form the Rhos empire. While all this was a very nice experience, the downside was that, once the Mongols arrived, the Golden Horde was stuck with 5-6 territories and they wouldn't move with their 130k+ army.
Just to test it, I took control over the Golden Horde khan and changed the Khazar ruler's religion to Sunni. Needless to say a CB popped up right away, whereas it was not available while the Khazar ruler was Tengri.
Imo this "Mongols getting stuck" factor somewhat diminishes the great impact their invasion has on Europe, and in my case spelled "game over" as it felt like there was nothing left to do.