For the Arch, the wording of the offering and the wording of the description seem to allude to different things. The offering says "without moving" which to me would suggest not specifically using the Move action of the middle of the turn, not using a potion that allows moving a hero etc. The description, though, says "must stay in their space" which would mean using the Spymaster banner action that doesn't say "move" and instead says "place your hero on a space" wouldn't cause the offering, but doesn't sound to me like it should be considered a "move" I think that could use some clarification.