It's possible to use ranged hunting bird attacks while on a horse.
This does not cancel the horse, does not do the normal attack, but does send out the bird.

Have not tested if this works with the melee versions of the hunting bird attacks, but all the attacks marked as Falconer can be used this way.
Thrill can also be used, but since it doesn't send out a bird and the normal attack is skipped, all it does is put the ability on cooldown.

Done on a male Shifter if that makes any difference.

I'll admit it's pretty fun to send a Death From Above at foes while running full speed ahead, but I don't think that's intended.