When players ride their horses or any other ridable entity, it will work for a little bit like it is supposed to, and then on the other players screens they will see each other standing still while their horses are riding.
Linked issues
Comments 3
I have tried on several servers. 1 was spigot, 2 was normal minecraft. So far it happens in all servers including the vanilla stock software.
Try viewing this from third-person mode (F5, I believe). If you don't see this happening on your side, yet others are observing it happening, it is most likely a server error.