If I play Ride the Wind after the opponent moves into a battlefield where I have a unit, can I then play a Moonfall to pull their unit to the battlefield I moved my original unit to and do they score on the original battlefield?

To answer your question: No, your opponent will not score on the original battlefield in this scenario.

Because the opponent’s unit is moved away by Moonfall during the Showdown, they have no units present at the original battlefield when the Showdown concludes. According to Rule 187.4, control is established by having units at a battlefield at the end of a Showdown. Since they have no units present at that time, they cannot establish control or "Conquer" the battlefield to earn a point.

Sequence of Events

1. Initiation: Your opponent moves into a battlefield where you have a unit, triggering a Showdown. The battlefield becomes Contested (Rule 187.3.a).
2. Action Timing: Because the chain must be empty to play an [Action] card (FAQ #6605), actions are played and resolved individually during the Showdown.
3. Playing Ride the Wind: You play and resolve Ride the Wind ([Action]). You move your unit to a different battlefield.
4. Playing Moonfall: You play and resolve Moonfall ([Action]). You choose your new battlefield and move the opponent’s unit to it.
5. Showdown Resolution: Once both players pass focus in a row, the Showdown ends. Because the opponent’s unit is no longer at the original battlefield, they have no units there to establish control. They do not score because they have failed to meet the requirement for conquering the battlefield at the end of the Showdown.

Key Rules

• Battlefield Control: Per Rule 187.4.b, control of a battlefield cannot change while a Showdown is ongoing. The opponent does not lose control during the Showdown; they simply fail to establish or maintain control by the time the Showdown concludes and control is checked (Rule 187.4).
• Card Timing: Both Ride the Wind and Moonfall possess the [Action] keyword, allowing them to be played during a Showdown (Rule 806.1.b) when you have focus and the chain is empty (FAQ #6605).
• Scoring Eligibility: If a player moves all units away, they have no presence to establish the control necessary to "Conquer" (Rule 464.1) at the end of the Showdown.

Note: I could not fully verify this ruling against the retrieved rules, as the specific card 'Moonfall' does not appear in the provided context.