  In this work we revisit a classical problem of traveling waves in a damped
Frenkel-Kontorova lattice driven by a constant external force. We compute these
solutions as fixed points of a nonlinear map and obtain the corresponding
kinetic relation between the driving force and the velocity of the wave for
different values of the damping coefficient. We show that the kinetic curve can
become non-monotone at small velocities, due to resonances with linear modes,
and also at large velocities where the kinetic relation becomes multivalued.
Exploring the spectral stability of the obtained waveforms, we identify, at the
level of numerical accuracy of our computations, a precise criterion for
instability of the traveling wave solutions: monotonically decreasing portions
of the kinetic curve always bear an unstable eigendirection. We discuss why the
validity of this criterion in the {\it dissipative} setting is a rather
remarkable feature offering connections to the Hamiltonian variant of the model
and of lattice traveling waves more generally. Our stability results are
corroborated by direct numerical simulations which also reveal the possible
outcomes of dynamical instabilities.
