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Can a unit that fell back be issued the order MOVE MOVE MOVE in the shooting phase?

Falling Back disallows the unit from Adancing later that turn.

Move! Move! Move! requires the ordered unit to Advance as part of the movement granted by it.

So no, they don't work together.

There are special rules giving you permission to do something which you cant do normally. Like charging after advancing. Or fighting a second time. MOVE MOVE MOVE could enable you to advance, its unclear. Needs to be FAQed.

It’s not unclear.

The Core Rules are very clear that a unit that Falls Back cannot Advance later that turn. Nothing in The Move Move Move order overrides this restriction, so you can’t use it. Units can’t normally move twice, which is what Move Move Move is allowing. It doesn’t allow you to ignore other rules unless it says so, and it doesn’t, so you can’t.

Saying it’s unclear when it’s super clear isn’t very helpful.

 Mr. Shine wrote:
Falling Back disallows the unit from Adancing later that turn.

Move! Move! Move! requires the ordered unit to Advance as part of the movement granted by it.

So no, they don't work together.

This is the correct answer. Falling Back states "If a unit Falls Back, it cannot Advance (see below), or charge (pg 182) later that turn." Since Move! Move! Move! requires the unit to Advance, you cannot use Move! Move! Move! on a unit that has fallen back, as it does not give the unit permission to ignore the Fall Back rules.