The Equipment attached to Avacynian Missionaries remains attached after it transforms into Lunarch Inquisitors.

Auras attached to the exiled creature will be put into their owners' graveyards. Equipment attached to the exiled creature will become unattached and remain on the battlefield. Any counters on the exiled creature will cease to exist.

If Lunarch Inquisitors somehow transforms into Avacynian Missionaries, the creature will remain exiled. If Avacynian Missionaries later leaves the battlefield, the creature will return to the battlefield.

In a multiplayer game, if Lunarch Inquisitors's owner leaves the game, the exiled card will return to the battlefield. Because the one-shot effect that returns the card isn't an ability that goes on the stack, it won't cease to exist along with the leaving player's spells and abilities on the stack.

If Lunarch Inquisitors leaves the battlefield before its triggered ability resolves, the target creature won't be exiled.

If a creature token is exiled, it ceases to exist. It won't be returned to the battlefield.