Can my opponent play an action to kill a unit I move into an objective I already control?

Ruling: No, your opponent cannot play Action spells/abilities during your turn unless a Showdown occurs. A Showdown only occurs when control of a battlefield would change, so moving your unit to an objective you already control does not trigger a Showdown.

Sequence:

• You control an objective with a unit already there
• You move another unit to the same objective
• Control does not change, so no Showdown occurs
• Your opponent cannot play Actions during your turn without a Showdown

Nuances:

• Your opponent can still play Reactions if you start a chain during your turn
• Moving itself does not start a chain
• If the unit being moved has an effect that triggers when something moves, it will open a chain