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.