Something I noticed is that a fair amount of famous fantasy fiction depicts dwarves riding griffins (e.g. Warhammer, Warcraft). Ignoring for a moment why dwarves seem to have little difficulty operating outside of their typical subterranean habitat, why domesticate griffins in particular? Is it because, according to real medieval bestiaries, griffin sniff out precious metals like gold and line their nests with it?
Maybe the griffins build their nests in mountain regions that can't be reached by climbing or even if you'd reach them you could hardly defend against them if you are hanging in a wall at 4000 meters.
So dwarfs could build access tunnels to these nests and slowly domesticate them. The first griffins could have been discovered when the dwarves carved a watch tower out of the rocks. The guards would have fed them and in return the griffins would live near the towers. And one time a brave (and probably a bit stupid) dwarf would have tried to ride one of them.
Large flying animals typically need to drop from high altitudes in order to take off. Some theorize that this is how Quetzalcoatlus got off the ground.
Thus, these griffins would probably live in mountainous areas, just like dwarves. That way, you have a flying animal strong enough to carry a dwarf that lives locally and is less expensive than a mechanical aircraft (If the dwarves are as advanced as that).
Okay, think for a minute about horses. We domesticated them thousand of years ago because of their ability to travel vast distances many times faster than we can. Griffens, and or peagasi, have the advantage of being able to fly, which is an amazing trait to harness for any species that wishes to travel, trade, or go to war. Unless there's significantly better choices in a very close proximety, there's no reason not to tame griffens.