Well we don't know yet,
However if you consider Hawking Radiation which shows that quantum effects allow black hole to emit black body radiation it can be theorized that Black holes and their event horizons can move.
And if it is correct that black holes are collapsed neutron stars, and being as we know stars move, this is further evidence that black holes do move.
Additionally if indeed galaxies move, and black holes are found in the centers of galaxies even more evidence for consideration.
I don't think you can assume that a Black hole will emit photons just because it moves, it's gravity field will move with it, still holding the photons.
As black holes are most likely moving and we do not observe photon radiation, this would seem to indicate that there is no "photon comet tail" resulting from black hole movement, or we would have seen it by now.
Suppose two black holes are moving towards each other on a collision course and one is larger than the other, you might think that the larger one could pull photons away from the smaller one. Not likely, the collision will happen and still no photons escape, you will just have a larger moving black hole. A feather and a stone fall at the same speed in a vacuum.
Now if a giant black hole is a spining moving collapsed neutron star then, that "comet tail" may be observed as the spiral arms of a galaxy.