This release changes how the teleportation is actually handled. Now, the method which does the teleporting accepts the variable for whether or not the entity was leashed.
If it was leashed before the teleport, we remove the leash holder, teleport it, and reattach the leash. Do note that this seems to have some visual bugs, so a relog might be necessary! The mobs seem to get stuck in mid-air for the client that does the teleporting, but appear fine for surrounding players.
If the animal was not leashed, we check if it is an entity that can sit (i.e. Parrots, Wolves, Cats, etc). If it's tamed, and the owner is the teleporting player, we teleport the entity. Since it wasn't leashed, we don't worry about the leash at all. Do note, if the sitting entity was leashed, it'll be teleported based on the method above first!