If a base scores and a minion moves to another base, like First Mate or Flying Monkey, and the new base has more power than the breaking point, does the new base score on this turn or next turn?
According to page 6 of the rules, the last thing you do after scoring a base (after discarding minions, and moving the First Mate and its ilk) is to "Check to see if another base is ready to be scored. Score it too, the same way."
So every time you score a base, you check all of the other bases and see if they're ready to be scored. The player whose turn it is still decides what order to score them in.