If enemy held at battlefield A, moved his unit to battlefield B, then they used another unit to move to battlefield A again, can I react to that?

Yes — you get a window to act, but not directly in response to the move itself. A Standard Move never opens a chain (Rule 441.3), so you can't play a Reaction "on" the move as it happens.

However, once the enemy moved their unit out of A, they lost control of it in the following cleanup, so their second unit moving into A applies Contested and stages a Non-Combat Showdown. During that showdown the enemy has Focus first, but when they pass — or after any chain they start resolves — Focus passes around, and you can play Action or Reaction cards (e.g., kill or bounce their unit) before it re-establishes control. They won't re-score A this turn since it was already scored, but the interference window is still there. If the moving unit has a "when I move" trigger, that trigger also opens a chain you can react to (FAQ #3218).