  We consider the dynamics of the Bose-Hubbard model on general lattices and
prove a Lieb-Robinson bound for observables whose supports are separated by an
initially almost particle-free region. We further obtain a maximal velocity
bound for particle transport through an initially empty region which also
applies to long-range hopping. Our techniques originate in the proofs of
maximal velocity bounds for Schr\"odinger operators and scattering theory in
non-relativistic QED.
