So, it is a misnomer that training in the Neutral Buoyancy Laboratory is intended for use in training or conditioning astronauts for freefall (“zero gee”) conditions. The purpose of the facility is really to allow astronauts to experience and practice extravehicular activities (EVA) in their bulky and awkward Extravehicular Mobility Unit (EMU), the A7L-derived suit, which at nearly 145 kg for the current suits is far too heavy to just walk around in on the surface of Earth. The neutral buoyancy provided by the pool helps the astronaut relieve that weight of the suit, and (with the assistance of divers and buoyancy control bags) kind of simulate being in different positions and orientations while trying to manipulate tools and hardware, but although it is possible to become disoriented in water it really doesn’t provide the experience of being in freefall and vacuum of space, and in particular the way that any action that applies a force results in a sudden and unexpected rotation around center of mass because it is retarded by the drag of the water in the pool.
For training in simulated freefall conditions astronauts do many sessions in the sometimes hated ‘Vomit Comet’, flying ballistic trajectories that give a few tens of seconds of genuine weightlessness at a time. I had the opportunity to fly on a commercial freefall flight simulator (I got to slot in for someone who had to bail out for a nominal medical exam fee), and while I have not been in the NBL I have extensive experience in scuba diving including occluded and restricted overhead (wreck diving) environments and can attest to how different those two experiences are.
As for a horse in the ~1/6 gee environment, it is going to have some severe problems compared to humans. A quadruped with digitigrade locomotion, it is evolved to most efficiently move at a trot and to sprint for relatively brief periods to escape predators, and it stops by essentially digging its front hooves into the ground. It is not evolved to shuffle along or balance delicately on the frog of the foot the way humans balance and pivot on the ball (their equivalent) or to hop along. Even worse, a horse has an enormous amount of mass with a relatively high center of gravity, and even though the downward acceleration (and thus, the reaction force) is much less on the Moon the inertia will be the same. That means that even if the horse can get enough traction to get up to a trotting speed, when they try to stop they are pretty much going to plow forward and potentially even go end over end if they can get traction on the front feet. Horses, of course, are pretty skittish animals that have a tendency to panic in unfamiliar conditions, and while falling over in Lunar gravity wouldn’t do the kind of damage that it might do on Earth, the lack of traction and unaccustomed mechanics would probably make it just as difficult to get back up. A donkey might adapt slightly better, although frankly if you plop a donkey onto the Lunar surface it is probably just going to stand there and refuse to budge until the gravity returns to normal because they are nothing if not obstinate in response to unscheduled changes in their lifestyle.
If you notice videos of NASA astronauts on the Moon, they are mostly hopping because that turned out to be the most efficient way to movie around. Some astronauts on the later missions did manage to adopt a “big step” kind of gate but that turned out to be problematic and in one case resulted in Jack Schmidt faceplanting with concerns that he might damage the helmet or get stuck on his side. Admittedly they were in those bulky A7L suits, not the shirtsleeves environment posited by the o.p., but the point remains that even humans have difficulty in adapting to the reduced gravity environment and we are basically evolved to be extremely efficient and adaptable walkers.
If you were going to try to adapt a quadruped to walk on Lunar gravity, an elephant would probably be your best choice given their plantigrade motion and maneuverability. It is left as an exercise for the reader as to how you would get an elephant to the Moon, what you would feed it when you got it there, and how you would deal with the resulting mounds of waste.
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