  We study the asymptotic behavior of a discrete-in-time minimizing movement
scheme for square lattice interfaces when both the lattice spacing and the time
step vanish. The motion is assumed to be driven by minimization of a weighted
random perimeter functional with an additional deterministic dissipation term.
We consider rectangular initial sets and lower order random perturbations of
the perimeter functional. In case of stationary, $\alpha$-mixing perturbations
we prove a stochastic homogenization result for the interface velocity. We also
provide an example which indicates that stationary, ergodic perturbations do
not yield a spatially homogenized limit velocity for this minimizing movement
scheme.
