I know there was a change to stop wild horses following you while holding wheat. However, tame horses only follow on the flat e.g. When crossing a river the horse used to follow you to dry land but now they remain in the water and will not take a step up to dry land. Will not follow you on dry land either. Cows and sheep will follow you up and down dale but horses no - they remain on the same level. You have to ride them to make them change level therefore as in river scenario you have to mount them in water.
Edit by [MCPE Helper] Zeb:
Tamed horses, donkeys, & mules WILL follow you when you are holding food, but they will not walk or jump up blocks when following you. This is both different from the behavior of farm animals, and different from Java edition behavior. The tamed horses/donkeys/mules should walk up 1-block-tall heights without any problem, especially since they can scale that distance without even jumping, and do so when wandering.