Calling the following to set an inventory for a horse puts a saddle and armor in its inventory but it is not visibly shown on the client:

entity.getInventory().setContents(inventory.toItemStack());

I have to explicitly set the saddle and the armor for it to update on the client:

entity.getInventory().setSaddle(entity.getInventory().getSaddle());
entity.getInventory().setArmor(entity.getInventory().getArmor());