  In [3] a simple discrete scheme for the motion of screw dislocations toward
low energy configurations has been proposed. There, a formal limit of such a
scheme, as the lattice spacing and the time step tend to zero, has been
described. The limiting dynamics agrees with the maximal dissipation criterion
introduced in [8] and predicts motion along the glide directions of the
crystal. In this paper, we provide rigorous proofs of the results in [3], and
in particular of the passage from the discrete to the continuous dynamics. The
proofs are based on $\Gamma$-convergence techniques.
