Riding a horse and bagging it at the same time still teleports you to the location where bagged entities are held, other animals probably produce the same result.

This one might be slightly more complicated to fix, as there's no way of blocking the interaction between the horse and player unless I do some weird stuff with invisible entities,

Will investigate, but the solution might just be to disable mountable entities from being put into the mob bag

Fixed in v1.9.1 (for real this time)