From a distance black holes look like a combination of gravitational lens and X-ray source (if they have material in orbit, which they normally do). If they form a binary system with a star then they influence the motion of the star too. But I think your question is about how a black hole would appear to an observer close to it, when there is nothing else near the black hole.
One can always adopt the perspective in which the black hole is not moving, and then the usual discussions apply.
If instead one adopts the perspective of a frame where the black hole moves along and encounters an astronaut, the astronaut gets pulled in towards the horizon and then, from the perspective of a far off observer, both black hole and astronaut move along together, with the latter slowly approaching the horizon. Of course you could also argue that the far-off observer may consider that the horizon itself is still in the process of forming, but this does not affect what they detect at their instruments. What their instruments detect is that the astronaut has gone: all signals from the astronaut have become dim and red-shifted to undetectability.
In response to a request for further information
Effects such as time dilation at the horizon do not influence the motion overall of a black hole. You can picture the spacetime around the black hole as somewhat like those illustrations using an elastic membrane, only now the distortion is extreme. As the black hole moves along (relative to other objects) the spacetime distortion moves along with it. To see this more precisely, simply note that the region of spacetime far from the black hole is just ordinary flat spacetime, and all motion is relative. So to find what a moving black hole does, just take a stationary black hole and then adopt another reference frame. This is basically the same procedure as one might adopt for considering the motion (at constant velocity) of anything at all.