  We study how strongly correlated electrons on a dissipative lattice evolve
from equilibrium when driven by a constant electric field, focusing on the
extent of the linear regime and hysteretic non-linear effects at higher fields.
We access the non-equilibrium steady states, non-perturbatively in both the
field and the electronic interactions, by means of a non-equilibrium dynamical
mean-field theory in the Coulomb gauge. The linear response regime is limited
by Joule heating effects and breaks down at fields orders of magnitude smaller
than the quasi-particle energy scale. For large electronic interactions, strong
but experimentally accessible electric fields can induce a resistive switching
by driving the strongly correlated metal into a Mott insulator. Hysteretic
$I$-$V$ curves suggest that the non-equilibrium current is carried through a
spatially inhomogeneous metal-insulator mixed state.
