  Understanding how the dynamics of a given quantum system with many degrees of
freedom is altered by the presence of a generic perturbation is a notoriously
difficult question. Recent works predict that, in the overwhelming majority of
cases, the unperturbed dynamics is just damped by a simple function, e.g.,
exponentially as expected from Fermi's Golden Rule. While these predictions
rely on random-matrix arguments and typicality, they can only be verified for a
specific physical situation by comparing to the actual solution or measurement.
Crucially, it also remains unclear how frequent and under which conditions
counterexamples to the typical behavior occur. In this work, we discuss this
question from the perspective of projection-operator techniques, where
exponential damping occurs in the interaction picture but not necessarily in
the Schr\"odinger picture. We suggest that a nontrivial damping in the
Schr\"odinger picture can emerge if the dynamics in the unperturbed system
possesses rich features. This suggestion is substantiated by large-scale
numerical simulations of charge transport in the extended Fermi-Hubbard chain,
where the nearest-neighbor interactions are treated as a perturbation to the
integrable reference system.
