  We consider dynamical transport metrics for probability measures on
discretisations of a bounded convex domain in $\mathbb{R}^d$. These metrics are
natural discrete counterparts to the Kantorovich metric $\mathbb{W}_2$, defined
using a Benamou-Brenier type formula. Under mild assumptions we prove an
asymptotic upper bound for the discrete transport metric
$\mathcal{W}_{\mathcal{T}}$ in terms of $\mathbb{W}_2$, as the size of the mesh
$\mathcal{T}$ tends to $0$. However, we show that the corresponding lower bound
may fail in general, even on certain one-dimensional and symmetric
two-dimensional meshes. In addition, we show that the asymptotic lower bound
holds under an isotropy assumption on the mesh, which turns out to be
essentially necessary. This assumption is satisfied, e.g., for tilings by
convex regular polygons, and it implies Gromov-Hausdorff convergence of the
transport metric.
