Perhaps the question could be better formulated in terms of a rescue mission? If someone or something falls into a black hole, and then an Alcubierre ship comes along, could they take a path that would intersect with the original falling object, and then still leave to a distance?
This formulation still needs refining, though, because an Alcubierre ship can, so far as we can tell, travel through time, so it could trivially just stay far away from the black hole and “rescue” the falling victim before they ever fell in in the first place.
