Ah, Hunter's pets, I'm surprised we put our stalwart companions into anyone else's hands. Our pets are our companions and those trusted to keep them safe, healthy and fed are the zoo keepers and trainers of Azeroth.
A flight master is an specialized trainer who provides transportation to those who cannot yet fly their own mounts of who simply want to enjoy the ride between cities. I wonder if the Wind Riders or Gryphons return to their home after each flight or if it is more like a one-way car rental where they stay at their destination until needed to return home or go elsewhere. They can usually be found at a roost or nest of their flying companions, which are usually bats, gryphons, hippogryphs, wyverns, or dragonhawks — depending (somewhat) on the race of the flight master.
Trainers and Breeders
Our mounts and companions have to come from somewhere. Us hunters can just go out into the wild and tame our own battle companions, training and shaping them into lifelong frien- oooh! a rare spawn! <abandon pet>... We can't tame out own mounts (yet - come on Blizzard, please) so we look to those with the skills and breeding stock to get us around Azeroth or for a cute cuddly kitten who loves to snuggle.