...how is it handled?
You are not even moving when you are mounted. The mount is the one moving so you can't tumble.
The Mount can tumble. I would personally rule you need a ride check to stay on at that point. No RAW to support it, though. Wraithstrike is right.
I suppose it'd just be handled normally. Maybe you'd suffer a penalty for "shaky ground." Certainly helps if you stand on the saddle (or is that a trick only Halfing Outriders can do?) in order to backflip or somersault to avoid an AoO.
You couldn't tumble through an enemy's space w/o the mount also doing so, though. Unless you were ok with dismounting in the process...