If my opponent pushes into my battlefield and I play Ride the Wind to move to the other battlefield, does the other combat resolve, healing units before starting the 2nd Showdown?

Yes, that is correct. According to the Riftbound rules and the relevant FAQ, when you use Ride the Wind to move a unit from one battlefield to another during combat, the combat at the first battlefield ends, and the unit fully heals before entering the new battlefield.

Here is the sequence of events based on the official rulings:

1. Combat Ends at Battlefield A: When you play Ride the Wind and move your unit, the combat at the original battlefield (Battlefield A) concludes.
2. Healing: As part of the transition, the unit fully heals from any damage it took during the combat at Battlefield A.
3. New Battlefield: The unit then enters the new battlefield (Battlefield B).
4. Second Showdown: If the conditions for a showdown are met at Battlefield B (e.g., it is contested), a new showdown will be staged there. This showdown will only begin after the first showdown has fully concluded.

Important Note: Per the RiftJudge FAQ (ID 2929), the combat in the first battlefield ends, the unit fully heals, and then it enters the new battlefield. If you are moving to a battlefield where a combat is already staged, the showdown at the new location will remain on hold until the original showdown at Battlefield A has completely resolved.