When a mob is inside of a climbing block (ladder, vines, twisted vines, crimson vines) they can no longer be pushed. I tried this with a villager, a sheep and a cow. All three didn't move. I only tried to push with the player, I do not yet know if this would be the case for other mobs pushing.

I got this bug easily reproduced by placing a mob near twisted vines (easy to hit to climb) and push him around for a bit. Sooner or later, the mob will consider himself climbing.