  Motivated by recent experiments with ultra-cold matter, we derive a new bound
on the propagation of information in $D$-dimensional lattice models exhibiting
$1/r^{\alpha}$ interactions with $\alpha>D$. The bound contains two terms: One
accounts for the short-ranged part of the interactions, giving rise to a
bounded velocity and reflecting the persistence of locality out to intermediate
distances, while the other contributes a power-law decay at longer distances.
We demonstrate that these two contributions not only bound but, except at long
times, \emph{qualitatively reproduce} the short- and long-distance dynamical
behavior following a local quench in an $XY$ chain and a transverse-field Ising
chain. In addition to describing dynamics in numerous intractable long-range
interacting lattice models, our results can be experimentally verified in a
variety of ultracold-atomic and solid-state systems.
