True, foot-based missiles beat mounted missiles hands down. If you're faced with many many horse archers, for example because you've become embroiled in the near east at some point, just bring lots of crossbowmen and enjoy the carnage.
As others and myself have pointed out before, all you need to win against the Mongols is at least even terrain (just withdraw if the battle map favors the Mongols), a solid melee infantry core (coz the main danger in fighting the Mongols lies in their lancers and the ensuing morale problems), and many crossbowmen. The infantry is just there to withstand sudden charges and give your missiles an anchoring point. The actual killing is done by thousands of crossbow bolts, or bodkin arrows or whatever you have.
This infantry+missile spam works well against many kinds of AI army, but is best if they rely on mounted missiles. You'll see the advantages of this vs. the Hungarians, Russians, and Byzantines to some degree, but against the Turks and Mongols, this is actually superior to any other force composition you might try - at least in my experience.
Timurids are a bit more difficult due to their War Elephants and excellent hybrid infantry (Sabardar!). If you can pick your terrain, though, it'll be OK, too. It is possible to take out those Elephants with conventional missile fire. I've even found crossbow bolts to work better than ignited arrows due to armor piercing.

Using these tactics, I'm actually glad when the AI brings lots of mounted archers: those are the least dangerous troops at its disposal!