  This paper continues the investigation of `Wasserstein-like' transportation
distances for probability measures on discrete sets. We prove that the discrete
transportation metrics on the d-dimensional discrete torus with mesh size 1/N
converge, when $N\to\infty$, to the standard 2-Wasserstein distance W_2 on the
continuous torus in the sense of Gromov-Hausdorff. This is the first
convergence result for the recently developed discrete transportation metrics.
The result shows the compatibility between these metrics and the
well-established 2-Wasserstein metric.
