If I control battlefield A with one unit during opponent's turn, opponent moves my creature to battlefield B with Charm, and in the same showdown I use Ride the Wind to move my unit back to battlefield A, do I score a conquer?

Ruling: Yes, you score a conquer at battlefield A.

Sequence:

• During the cleanup after the Charm (where the combat/showdown are staged at B), control of battlefield A is cleared
• When you move back to battlefield A with Ride the Wind, you contest it again because you no longer control it
• After the showdown/combat at B is resolved, a showdown will start at A
• If your unit isn't removed from battlefield A during that second showdown, you will conquer it