How could I modulate smoothly from A# minor to B minor?
When modulating by a half step, one trick is to exploit the enharmonic equivalence between the German augmented-sixth chord and the dominant-seventh chord.
In A♯ minor, the German augmented-sixth chord is
F♯ A♯ C♯ D𝄪; since this chord sounds the same as the V7 in B (
F♯ A♯ C♯ E), you can easily resolve to B minor and strengthen the modulation from there.
This also works going backwards: you can go to a V7 in B minor, but then resolve it as if it were the German augmented-sixth in A♯ minor.
Modulation and key change generally speaking uses the V chord of the new key. So you'd have a bar or two of F♯ or F♯7 leading into the new key of Bm.
You could also lead into that F♯ with an A♯diminished chord, using it as a pivot note. That gets the job done in one or two bars.