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I'd imagine that the neutron star would go into an orbit around the black hole and that the black hole would slowly pull material off the neutron star. So this would mean that the neutron star would be all around the black hole so the overall effect would keep the black hole virtually in the same place. Though it would depend on the ratio of the masses between the two objects. However, I'm not sure how a neutron star would react to itselt being pulled apart since its just degenerate neutrons.
Even if the black hole's velocity changed there would still be no information left over about the neutron star. You wouldn't be able to tell what caused the black hole's velocity to change.
I'm quite possibly just blowing a lot of smoke and making no real sense and/or just being completely wrong. Either way, its still interesting stuff.
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