  We study the response of a highly-excited time dependent quantum many-body
state to a sudden local perturbation, a sort of orthogonality catastrophe
problem in a transient non-equilibrium environment. To this extent we consider,
as key quantity, the overlap between time dependent wave-functions, that we
write in terms of a novel two-time correlator generalizing the standard
Loschmidt Echo. We discuss its physical meaning, general properties, and its
connection with experimentally measurable quantities probed through
non-equilibrium Ramsey interferometry schemes. Then we present explicit
calculations for a one dimensional interacting Fermi system brought out of
equilibrium by a sudden change of the interaction, and perturbed by the
switching on of a local static potential. We show that different scattering
processes give rise to remarkably different behaviors at long times, quite
opposite from the equilibrium situation. In particular, while the forward
scattering contribution retains its power law structure even in the presence of
a large non-equilibrium perturbation, with an exponent that is strongly
affected by the transient nature of the bath, the backscattering term is a
source of non-linearity which generates an exponential decay in time of the
Loschmidt Echo, reminiscent of an effective thermal behavior.
