  We explore the stability of far-from-equilibrium metastable states of a
three-dimensional Coulomb glass at zero temperature by studying charge
avalanches triggered by a slowly varying external electric field. Surprisingly,
we identify a sharply defined dynamical ("depinning") phase transition from
stationary to nonstationary charge displacement at a critical value of the
external electric field. Using particle-conserving dynamics, scale-free
system-spanning avalanches are observed only at the critical field. We show
that the qualitative features of this depinning transition are completely
different for an equivalent short-range model, highlighting the key importance
of long-range interactions for nonequilibrium dynamics of Coulomb glasses.
