If an effect moves pieces "on each board," "in each land," or for "each Spirit," can the same piece be moved multiple times?

If an effect moves pieces "on each board," "in each land," or for "each Spirit," can the same piece be moved multiple times?

Yes, assuming that it separately meets the condition to be moved each time. For example, "Beasts Chase Off Homesteaders" ("On Each Board: Push up to 2 Explorers or 1 Town from a land with Beasts tokens.") can Push a Town multiple times, so long as the Town keeps arriving in a land with Beasts tokens that's on a board where "Beasts Chase Off Homesteaders" hasn't resolved yet.
Note however that while an "each [piece]" Event/Fear/Adversary effect does create one action per land with that piece, each piece of that type only participates in one of those actions. For example, while "Beasts Prowl" ("Each Beasts token generates 1 Fear if Invaders are present, and moves to an adjacent land if not.") can move a Beasts token into a land that already has Beasts tokens, the newly-arrived Beasts token doesn't participate in the "Beasts Prowl" action there – only count the Beasts that were already there before resolving "Beasts Prowl."