Does moving an enemy unit to a different battlefield cause a showdown?

Moving an enemy unit to a different battlefield causes a showdown only if that move makes the battlefield contested.

A battlefield becomes contested when a unit moves to a location not controlled by that unit's controller (Rule 445). Therefore, a showdown occurs if the unit moves to an uncontrolled battlefield or a battlefield controlled by a different player, but it does not occur if the unit moves to a battlefield already controlled by its own controller (FAQs 4407, 8066, 5980).