I've just found another convenient way to do this.

1. Make sure the clip position/data isn't locked (It doesn't work if locked).
2. In the Piano Roll view, select the notes.
3. Press the numpad key "5." This will open Preferences.
4. Do nothing and close Preferences.
3/4. Close the Piano Roll view.
5. Press the numpad key "2" to move the notes to the next track, or "8" to the previous track.

I don't know why this works, though :) 

Edit: Sorry, Step 3 & 4 were unnecessary. Just close the Piano Roll view and do Step 5.