Ninja Dojo had first mate played on it, does the winner get to choose which minion to destroy before or after first mate moves. Both of those cards happen after the base is scored and the rules say:
If multiple effects happen at the same time,
the player whose turn it is determines
And I'm not sure of the particulars of your situation but If you're wondering if First Mate can escape destruction by moving the answer is he can't. Ninja Dojo says the winner may destroy any one minion not any one minion at this base.