Characters using this ATA can be given the free move action at different points during the beginning of the turn, right?

Example: GCPD Motorcycle is given a free move action and carries Harvey Bullock and Commissioner Gordon. Bullock and Gordon are then each given a free move action using the ATA.

Correct, they all occur at the same time. So the active player(you) gets to choose in what order they unfold in. As long as its all before you give anyone a non free action.

If by "they" you mean all the potential "at the beginning of the turn" free actions, then you are correct. I'm just trying to specify that you do not have to activate the ATA on all figures at the same time. You could move one figure with the ATA, outwit with another, poison with another, then move another with the ATA, and so on.