is it before the ship moves, after the ship moves, or both?

Once per round, after a friendly ship at Range 1-2 executes a white maneuver, you may remove 1 stress token from that ship.

It would be after the ship moves. You haven't executed the maneuver until after you place the ship at the far end of the template.

You measure after the friendly ship executes a maneuver.

A stressed ship executes a maneuver that brings it within range 1-2 of the ship Kanan is on. You do the white maneuver, remove a stress, perform an action if now unstressed.

I can see the reason for the question here. I would normally read after X does Y to mean X describes the state before Y happens, not after.

It's a friendly ship at range 1-2, and then it executes a maneuver.