Using the finite volume method, one can define a discrete Kantorovich distance with a Riemannian structure based on a Euclidean mesh. We show that in most cases, the limit distance as mesh size tends to zero, in the sense of Gamma- or Gromov-Hausdorff-convergence, is strictly less than the standard Kantorovich distance. This is due to an oscillation effect reminiscent of homogenization. We introduce a geometric condition on the mesh that prevents oscillations and are able to show Gromov-Hausdorff convergence under this condition.