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After reading the rules again it's definitely not obvious, but I think using other action cards as the basis for a decision gives us the answer.
For example, one of the action cards is to add another castle block. It wouldn't make sense if this wasn't a free action, or you'd just use an action to do it. So this card clearly gives you something for free. Essentially you've "prepaid" by spending an action to take the card in the first place.
Given this I'd say that you do not pay an ap for the movement of the knight in your example above.
That's good reasoning and that makes sense to me now. Thanks for pointing that out!