If an opponent controls both battlefields at the start of my turn, and I kill their unit on one battlefield, then they move their unit from the other battlefield during my attack, do they still score a point even though they controlled that battlefield at the start of my turn?

Ruling: Yes, they score a point. You can only score a battlefield once per turn, and since they haven't scored it during your turn yet, they can score it when they gain control at the end of the resulting showdown.

Sequence:

• Opponent controls both battlefields at start of your turn
• You kill their unit on battlefield A
• You attack battlefield B (where they have a unit)
• Opponent moves their unit from battlefield B to battlefield A
• A showdown occurs at battlefield A after your current combat ends
• If opponent is the only player controlling units at battlefield A at the end of that showdown, they score a point

Nuances:

• To score a battlefield, you must not have scored it that turn AND you must gain control of it at the end of the showdown
• Moving to a battlefield you already control does not open a showdown
• The showdown only takes place after the current combat ends