If I lose control of a battlefield on my opponent's turn and ride the wind a unit back on after they control, do I score?

Yes, you can score a point in this situation, provided you meet the specific conditions for conquering a battlefield.

According to FAQ #9040 and FAQ #2364, you can use Ride the Wind to move a unit to a battlefield, conquer it, and score a point, even if you already scored at that battlefield during your own turn. Scoring is tracked per-turn for each player.

However, there are critical requirements for this to work:

1. You must lose control first

As established in FAQ #1067 and FAQ #2140, you cannot score if you maintain control of the battlefield throughout the entire showdown. You must completely lose control of the battlefield (the opponent must establish control) and the initial showdown must end before you can score it again.

2. The "Conquer" requirement

To score via Ride the Wind during your opponent's turn, you must successfully establish control of a battlefield you did not control at the start of the combat/showdown.

The Sequence

1. Opponent's Turn: Your opponent moves a unit into the battlefield you previously controlled, triggering a showdown.
2. Loss of Control: You lose the showdown, and your opponent establishes control of the battlefield.
3. Your Opportunity: After that showdown finishes (both players pass focus in a row without playing an action), you play Ride the Wind to move your unit into that battlefield.
4. New Showdown: Because you moved a unit into a battlefield controlled by your opponent, a new showdown begins.
5. Resolution: If you win this new combat showdown, you establish control of the battlefield. Because the battlefield transitioned from being under your opponent's control to yours, you conquer it.
6. Scoring: Since you have not yet scored at that battlefield during this specific turn (the opponent's turn), you successfully score a point.

Important Nuance: If this would be the final point of the game, you cannot score it unless you have already scored at every battlefield "this" turn. If you have not, you would draw an extra card instead of gaining the final point (FAQ #7467).