I'm not sure that this is the best advice for dealing with the Mogols. The Timirids are a different story but I've either won or lost by the time they come so they are not that big an issue

Unlike MTW i cannot get a huge army in Asia Minor that can simply wait for the Mongols to appear and then wipe them out.

In M2TW you are unable to do that.

Trick #1 Let them take a worthless city. What is a worthless city? One that only has walls. When you see where the Mongols are going destroy everything in the city that can be destroyed This will force the Mongols to garrison the city after the siege it with troops that they have to pay for. If possible don't build up the agriculture on Eastern cities. However, make them place the city under siege. Have a 20 unit stack of Spear Militia and Archer Militia on the walls If possible put a family member in the city that has 2 or 3 command stars. Let the Mongols sit outside your walls for 10 years. This gives them a base but it also gives you time. You can have a fast army that goes and sieges the city every time they leave. When they move back to break the siege you retreat.

Trick #2 Lead them on a merry chase. This works better if the Mongols show up near Tbilisi. Have an army of cheap fast cavalry. They don't have to be good but they have to be fast. If you have a family member with the forced march attribute that makes them faster put him in the army as well. Then move this army where the Mongols can see it and let them come after it. However, you need to place your army so that the Mongols have to expend most of their movement to reach you. When they attack refuse combat and retreat. Then move to the limit of their range again. By doing this You can sometimes get them chase you all the way to Sarkell At that point they are the Russians, and Polish factions' problem. This is hard to do but it can be done and is quite fun if you are also at war with the Polish or Hungarians.

Trick #3 Assassinate their leader. If you build a force of about 20 assassins and have them practice on easy targets like Catholic priest and Venetian merchants they are then able to kill the Mongol leader. When the Mongols appear they don't have much in the way of faction leaders. Kill the leaders and the Mongols become bandits albeit very powerful bandits. However, their behavior changes