For Pet Event team slots (or any event team slot, for that matter) simply dragging troops on the battle prep screen will change the order for the next battle only, then revert. For your normal roster of teams (the set you can use in PvP, explore, quests, etc), if you change the order by simply dragging on the battle prep screen, it will save the order so long as you either play the battle or back out to the map (as opposed to just closing the game outright from this screen). As others have pointed out, to get the order to stick for event teams, you have to go into edit teams.
I’m not sure if either functionality is universally “better” for everyone (I personally prefer that the team order just stick always like with normal teams), but this inconsistency bothers me.