What happens to the fabric of space in the wake of a moving black hole? Is space permanently deformed by a moving black hole or does it rebound as the black hole passes?
A black hole creates curvature in spacetime just like any other massive object, such as a star, which makes it attract other matter gravitationally. A black hole isn't any more exotic than a star in that regard.
The black hole's gravitational field is just a consequence of its mass existing in that location. If a black hole is in motion relative to you, then the curvature of spacetime that it creates will follow it.
You could, of course, also argue that the black hole is at rest, and you're the one who's in motion. That's why it wouldn't make sense for a black hole to "permanently" deform spacetime, since that would mean that there's a preferred frame of reference in which the black hole is in motion.