Can I activate a Hidden card (Consult the Past) on battlefield A when my opponent moves to conquer battlefield B, or does the action need to be on my battlefield?

Ruling: You can flip any Hidden card whenever you could legally react, regardless of which battlefield it's on. Moving itself doesn't give you an opportunity to react, but moving to start a Showdown does. If your opponent plays an Action anywhere, you can react to that and flip any Hidden cards.

Sequence:

• Moving alone does not trigger a reaction opportunity
• Moving to start a Showdown (because the battlefield is controlled by an opponent or no-one) gives you something to react to
• Playing an Action anywhere gives you an opportunity to react

Nuances:

• Hidden spells and effects that target must target things at the battlefield they are played to (where it makes sense)
• Consult the Past doesn't target anything, so this restriction doesn't apply to it