To my mind, there are 2 aspects to actually scoring a base: breaking the base and scoring the base.
Breaking a base happens at the end of a player's turn. This happens when the total value of minions exceeds the base's limit.
Then players have the opportunity to play special cards, knowing that the base is going to give out points regardless. If you reduce the value of an opponents minions, kill some off, or move some away such that the minion score is below the value that it takes to break a base, it no longer matters because the base has broken already. The only thing that remains is to see who has the most, second most, and third most minions on the base and give them the appropriate number of points.
Breaking a base happens during phase 3 of a turn. After that you still have the draw cards phase and the end turn phase.
(see: the phases of a turn, or this is how you roll in the rulebook)
Once a base has been chosen to break (step 2 of the scoring order list in the more recent rulebooks), it doesn't stop anymore.