Even though I got this by deliberately aiming and shooting at the horses, later in the game I found an easier way:
First of all, this is a LOT
easier once you have a level 9 or 10 gun, especially the high damage ones.
While you're grinding for Without a Grudge you'll find that you bump into either police or gang carriages, this causes them to chase you - when they're near move the camera around to highlight the horse and press
, most of the time a high level gun will kill the horse in 1 shot, if not, take another - the Black Marias have 2 horses, so killing 1 won't make them flip, but the second one will.
Note that it's easier to keep whipping the horse to make the enemies come at you faster, thus more chances to get them to flip.
While I was grinding for Without a Grudge I ended up flipping about 20-30 of the enemy carriages, so you don't have to deliberately go for it if you intend to get all of the achievements.