  We consider the propagation of a single particle in a random chain, assisted
by the coupling to dispersive bosons. Time evolution treated with rate
equations for hopping between localized states reveals a qualitative difference
between dynamics due to noninteracting bosons and hard-core bosons. In the
first case the transient dynamics is subdiffusive, but multi-boson processes
allow for long-time normal diffusion, while hard-core effects suppress
multi-boson processes leading to persistent subdiffusive transport, consistent
with numerical results for a full many-body evolution. In contrast, analogous
study for a quasiperiodic potential reveals a stable long-time diffusion.
