Can the horizon of a black hole move? Because of time dilation we cannot observe a black hole forming in a finite amount of time. For the same reason I suppose we also cannot observe the horizon moving: everything happening on the horizon takes an eternity to witness from the outside perspective.
Therefore, would a moving black hole result in new horizons (almost) forming according to an outside observer? The initial horizon would remain frozen in time, followed by the horizons around the moving singularity. Meaning, would moving black holes leave a trail of blackness behind, everywhere it passed?
An important issue here is the moving reference frames. Could one really claim that there is a perspective where the observer moved, rather than the black hole? The observer isn't curving spacetime to extremes, while spacetime is a medium: it's a fabric, it seems more than something described by coordinate systems.
Edit:
I changed the question title, it used to be "Moving reference frame of a black hole" but the new title better suits my question. The issue with reference frames is more a follow-up question.
 A: GR doesn't have global reference frames, only local ones. Therefore you can't have a frame of reference big enough to surround a black hole.
So a better way to phrase this question would be in terms of a moving observer. No, a moving observer cannot observe the horizon. The definition of the horizon is that causal curves from the horizon cannot reach outside events, and this definition precludes any external observer from observing the horizon. This definition is independent of the state of motion of an observer, and is independent of any choice of coordinates.

Because of time dilation we cannot observe a black hole forming in a finite amount of time.

It's not really because of time dilation, it's simply because the definition of a horizon is that it's something you can't observe (the boundary of an externally unobservable region of spacetime).
A: Since I found an answer to my own question, which I first only wrote as a comment, I'll put it here to wrap things up:
Just like photons don't age but still move, black hole horizons don't age but still move.
