Purge the Land With Salt and Fire

When an effect does multiple things "on each board," can each thing happen in a different land?
Yes in principle, but most such effects specify otherwise.
By default, as long as you're resolving everything on a particular board, you can split the effect. For example, "On Each Board: Remove 1 Beasts. Add 1 Disease" (part of the first choice of the "Life's Balance Tilts" Event) can remove a Beasts token from one land and add a Disease token to a different land on the same board. And similarly, "On Each Board: Push up to 2 Dahan" (part of the first choice of the "Hard-Working Settlers" Event) can Push 2 Dahan from different lands on the same board.
Nevertheless, many of these effects specify that they happen in a single land on the board, which means that they can't be split. For example, if "On Each Board: Gather 1 or 2 Dahan into a land with Dahan Setup Symbols." (the "Return to Old Pillars" Dahan Event) moves two Dahan, it must move them into the same single land, since the Dahan are gathered into "a land" (singular).
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When an effect happens "on each board," can it be split among multiple lands on that board?
Yes, unless the effect affects "a land" / "a single land" or is otherwise more specific.
To illustrate the difference, "On Each Board: Push 2 Dahan from a land with Blight to a land without Blight" ("Forsake the Barren Land") must Push both Dahan from the same land because they must both leave "a land with Blight." In contrast, "On Each Board: Push up to 2 Dahan" ("Hard-Working Settlers") can simultaneously Push Dahan from different lands, since it doesn't set any conditions on which land(s) the Dahan leave.
Similarly, if such an action has multiple instructions, like "On Each Board: Add 1 Beasts. Remove 1 Disease." ("Life's Balance Tilts"), it can resolve each of its instructions in a different land (in this case, adding a Beasts token in one land and removing a Disease token from a different land).
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