  We analyze the propagation of correlations after a sudden interaction change
in a strongly interacting quantum system in contact with an environment. In
particular, we consider an interaction quench in the Bose-Hubbard model, deep
within the Mott-insulating phase, under the effect of dephasing. We observe
that dissipation effectively speeds up the propagation of single-particle
correlations while reducing their coherence. In contrast, for two-point density
correlations, the initial ballistic propagation regime gives way to diffusion
at intermediate times. Numerical simulations, based on a time-dependent matrix
product state algorithm, are supplemented by a quantitatively accurate
fermionic quasi-particle approach providing an intuitive description of the
initial dynamics in terms of holon and doublon excitations.
