  We rigorously show that a local spin system giving rise to a slow Hamiltonian
dynamics is stable against generic, even time-dependent, local perturbations.
The sum of these perturbations can cover a significant amount of the system's
size. The stability of the slow dynamics follows from proving that the
Lieb-Robinson bound for the dynamics of the total Hamiltonian is the sum of two
contributions: the Lieb-Robinson bound of the unperturbed dynamics and an
additional term coming from the Lieb-Robinson bound of the perturbations with
respect to the unperturbed Hamiltonian. Our results are particularly relevant
in the context of the study of the stability of Many-Body-Localized systems,
implying that if a so called ergodic region is present in the system, to spread
across a certain distance it takes a time proportional to the exponential of
such distance. The non-perturbative nature of our result allows us to develop a
dual description of the dynamics of a system. As a consequence we are able to
prove that the presence of a region of disorder in a ergodic system implies the
slowing down of the dynamics in the vicinity of that region.
