  We consider the dynamics of a system of free fermions on a 1D lattice in the
presence of a defect moving at constant velocity. The defect has the form of a
localized time-dependent variation of the chemical potential and induces at
long times a Non-Equilibrium Steady State (NESS), which spreads around the
defect. We present a general formulation which allows recasting the
time-dependent protocol in a scattering problem on a static potential. We
obtain a complete characterization of the NESS. In particular, we show a strong
dependence on the defect velocity and the existence of a sharp threshold when
such velocity exceeds the speed of sound. Beyond this value, the NESS is not
produced and remarkably the defect travels without significantly perturbing the
system. We present an exact solution for a $\delta-$like defect traveling with
an arbitrary velocity and we develop a semiclassical approximation which
provides accurate results for smooth defects.
