Minecraft wiki entry for Zombie_Horse, associated text: A ridden horse automatically runs up any one-block-high slope. The horse and rider can safely fit through a space as low as 2.75 blocks high. Lower clearance risks suffocating the rider if the rider's head enters a non-transparent block. The horse itself can enter gaps as low as 1.625 blocks high, but may itself take suffocation damage when clearance is less than 1.75 blocks. Horses cannot fit through a 1-block-wide gap.