  This paper deals with dynamical optimal transport metrics defined by spatial
discretisation of the Benamou--Benamou formula for the Kantorovich metric
$W_2$. Such metrics appear naturally in discretisations of $W_2$-gradient flow
formulations for dissipative PDE. However, it has recently been shown that
these metrics do not in general converge to $W_2$, unless strong geometric
constraints are imposed on the discrete mesh. In this paper we prove that, in a
$1$-dimensional periodic setting, discrete transport metrics converge to a
limiting transport metric with a non-trivial effective mobility. This mobility
depends sensitively on the geometry of the mesh and on the non-local mobility
at the discrete level. Our result quantifies to what extent discrete transport
can make use of microstructure in the mesh to reduce the cost of transport.
