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I was wondering how you can handle the situation when both players have a unit that has the ability 'at the start of the battle, but before the first turn begins, this unit can move X or deploy'.

Who goes first in this action? Is it the player who's turn it is? or one after the other, like regular deployment? or something else?

Both players roll a dice, the winner decides the order.

From the Rulebook FAQ:

Q: If both players have units with abilities that allow them to be set up during deployment ‘after both armies are fully deployed’ (e.g. Ratlings’ Find the Best Spot), which player deploys their units first?

A: The players should roll off and the winner decides.

From this you can figure out the same applies in your situation.