While riding a saddled horse and moving upwards both at normal speed and galloping/running speed(With shift by default) Horse will glitch and appear as it got stuck in permanent transition between those two blocks, although there is no change in height. Somehow after this glitch happens, horse will take damage when dismounting.

This glitch doesn't affect only horses, it also appears to affect donkeys, who die when dismounted due to effects of the glitch and their low HP. 

From observation it seems like horses/donkeys "teleport" within second and take damage when dismounting due to that glitch, also seems that more you do it the more damage horse takes(when mounted).

Apparently horse armor decreases the received damage to the horse(unclear though).

So far I haven't witnessed any effects on the player.

How to recreate a bug:

1. Tame a horse
2. Equip saddle on a horse
3. Ride towards elevated blocks
4. Do above few times and ride around for a minute and then dismount horse to recreate damage due to that glitch. 

Note: glitch seems like it appeared in 1.14 Pre-release 1 and is consistent in Pre-release 2. It is unclear what is the reason for the glitch, but it appears as if horse gets stuck while going up, and then because of that horse glitches and then "teleports" when dismounted and receives damage.