Can I react with Gust to my opponent entering a battlefield they already conquered?

Ruling: No, you cannot react to movements between two controlled locations. Movements between controlled locations do not create a reaction window.

Nuances:

• If a player conquered a battlefield, then left it (making it uncontrolled), and then moved back in, that would cause a showdown because control is changing from uncontrolled to controlled
• If a player has a unit in a battlefield they're controlling and moves another unit into it, that doesn't contest control so no showdown occurs
• A player's base is always considered controlled by its owner