Occasionally (frequently) horses on leads do not step up a block, the lead appears to stretch out even though it has actually been broken. I had hoped this would be fixed with the inability to lure mobs up blocks with food. However it is a separate bug (possibly related).
The horses sometimes step up normally but if you want to lead a horse a long distance with elevation changes then you are going to have a bad time. (Donkeys, mules) Occurs regardless if leashed horse has been tamed or not.
See attached screenshots. The horse appears to be on a lead in both, but it has already broken. You can see the broken lead on the ground in the second shot. I believe the inventory reduces by 1 when it breaks but the screenshot was taken in creative mode. I typically bring more than 1 in survival because it is very easy to lose them when they break so easily.