I've been trying to get a "flying taxi" datapack together and ran into an issue when trying to teleport floating horses and pig vertically while riding them.

To replicate:

Summon a no-gravity horse or pig with a saddle above the ground.

Ride the spawned mob.

Run a command to teleport the mob directly above or below its current position (i.e. /execute as @e[type=horse] at @s run tp @s ~ ~4 ~)

The ridden mob will seemingly not teleport with the player riding it. It seems to work mostly correctly with grounded pigs, and work occasionally with flying pigs, and occasionally work with horses when there's also a horizontal change in teleportation.

My expectation is that teleporting a mob, even when it's ridden, will teleport it and the ridden player to the chosen location every time, or leave the player in their position and teleport only the mob to the chosen location.