Really the only power that moves the AI is de jure, especially de jure capitals. The AI is absolutely fixated by the de jure capital of its primary title, going as far as revoking the county from its current holder and holding it itself, even if the AI ruler's primary demesne is somewhere else. Examples: Abbasid Caliph revokes Damascus, Sweden revokes Uppland, HRE revokes Jülich, Mamluk Sultan revokes Caliph's Cairo (and this is one of the most egregious examples. The Sultan revokes Cairo from the Caliph! The bastard granted him that title and now AI madness is going to revoke it!).