[thaumasiotes]

Time has also stopped for photons, which does nothing to stop them from moving from place to place.

I don't see a way to read your comment that allows for the possibility that a black hole might move, which is something they do.

I'm riiiiiiight on the edge of concluding that you are a poet and the only thing you know about the words is the way they sound. Is there more to it than that?

[GoblinSlayer]

Technically time is exponentially dilated, this introduces large relativity of synchronism: the same thing takes different time in different reference frames. You aren't completely wrong to say black holes will fly through each other, but this doesn't take the same time in all reference frames. For a distant observer it takes infinite time after their event horizons touch, and Hawking radiation evaporates everything before that.

How black holes can move is an interesting question, but as you can see, in reality there are no infinities, since stuff slows down before that. Maybe infinities could exist for an observer inside black hole, but an observer outside of black hole sees only large slow down.