A blackhole traveling at near light speed is pretty darned close to the analogy of a massive object instantaneously disappearing, similar to fictional spaceships engaging their warp engines.
Of course, it's not actually disappearing, just moving, but the original point was about detecting sharp changes in the gravitational waves. A quick Google search tells me that gravitational wave red shifting is a thing, and I imagine that with blackholes it's a very important phenomenon and area of study. And I would guess that there can also be interesting second-order effects that such a blackhole's movements have on the propagation of gravitational and electromagnetic waves from other objects.
> with blackholes it's a very important phenomenon
Black holes can lense gravitational radiation emitted by background systems.
Most background systems we are likely to detect soon will involve black holes. But these are black holes in some sort of mutual orbit, rather than black holes simply moving across some system of celestial coordinates.
For black holes that are moving linearly at near the speed of light, the black hole's effect on the metric elongates like a pencil, with the field weak outside and growing strong towards the centre of the "lead" or graphite. This is similar to Lorentz-contracting the near region around the black hole, and one can generalize a bit and say that as the boost between an observer and any object increases, the object thins. In the ultra-ultrarelativistic limit, the object and all the strengthening-towards-infinity field values around it become infinitely thin.
As one's speed relative to a black hole gets very close to c, the black hole becomes quite easy to model as an exceptionally high-energy massless particle.
You get this effect when your small space capsule whizzes by our galaxy's central black hole at speeds near that of light too, and your small momentary perturbation basically affects the black hole not at all. Because Lorentz contraction is reciprocal, whizzing a black hole -- even a large one -- at ultrarelativistic speeds past the International Space Station is going to have very little effect on it.
Tossing a large black hole past the ISS at low speeds compared to light will really mess up the neighbourhood of the solar system, but your space capsule can pretty safely manage a slow-compared-to-light hyperbolic orbit around a large black hole without much problem (ignoring any accretion disc and twin "paradox" issues).
Of course, it's not actually disappearing, just moving, but the original point was about detecting sharp changes in the gravitational waves. A quick Google search tells me that gravitational wave red shifting is a thing, and I imagine that with blackholes it's a very important phenomenon and area of study. And I would guess that there can also be interesting second-order effects that such a blackhole's movements have on the propagation of gravitational and electromagnetic waves from other objects.