When getting off of a tamed horse, donkey, or mule, there's a (seemingly random) chance that it'll take damage.

While riding a tamed horse, donkey, or mule, there's also a (seemingly random) chance that you'll teleport back a few blocks with it, causing both you and them to take damage or instantly die. The death message says "[player name] hit the ground too hard."

This glitch happened to my friend (who ended up going through 4 horses because of it) and I while playing on a LAN server and to me offline on my own. I'm not sure if there are specifics to replicating it; you'll probably get it just by riding a horse around a hilly area and dismounting it a few times though.