If I use Ride the Wind during my opponent's turn, and move from Battlefield A to Battlefield B, then he buffs his unit, and I decide to use Ride the Wind again, from Battlefield B to Battlefield A, do I score? Since the other battlefield was empty for a while?

No, you will not score a point at Battlefield A if you have already scored it earlier in the turn.

According to Rule 447 and FAQ #7507, a player may only score each battlefield once per turn, regardless of whether they leave and re-enter it. Scoring a battlefield earlier in the turn prevents you from scoring it again, even if you move away and return.

Sequence of Events:

1. Initial State: You control Battlefield A and have already scored it this turn.
2. Move: You use Ride the Wind to move your unit from Battlefield A to Battlefield B.
3. Opponent Action: Your opponent initiates a showdown at Battlefield A.
4. Return: You use Ride the Wind to move your unit from Battlefield B back to Battlefield A.
5. Scoring: Because you have already scored Battlefield A this turn, you cannot score it again, regardless of the outcome of the showdown.

Important Nuances:

• Separate Turns: As clarified in RiftJudge FAQ #2364 and FAQ #6080, the restriction of "once per battlefield per turn" applies to your turn and your opponent's turn separately. Scoring on your turn does not prevent you from scoring that same battlefield again on your opponent's turn.
• Control Requirement: You must lose control of the battlefield completely (it must be empty) between showdowns to be eligible to score it again (FAQ #1067).
• Showdown Resolution: Per FAQ #2099 and FAQ #4666, if you move to a battlefield that is currently part of a showdown, the showdown at the original location must conclude before the showdown at the new location can be fully resolved and scored.

This interpretation is based on the mechanics of battlefield control and scoring established in Rule 446.1 (Conquer), Rule 447, and the specific scenarios outlined in FAQ #7507, FAQ #2364, and FAQ #6080.

Note: I could not fully verify this ruling against the retrieved rules.