Crows are birds so they really shouldn't have to fear fall damage since they can fly, but it seems crows on your shoulder will just stay there regardless of what's going on, which is a problem since when a crow is riding you, for some reason they take the same amount of fall damage as you, and with their small health pools and that they can sneak up onto your shoulder without you noticing for a few seconds, it's really easy to accidentally kill your crows by jumping a few too many blocks down.

I just experienced this with my first tamed crow. It would be really nice if they would be forcibly dismounted instead of taking damage. I'm sure real birds would fly off of you if you were about to fall on the ground after all, and crows are super smart, so it's a shame seeing them die from something so easily preventable when they have wings.