The diffusive transport distance, a novel pseudo-metric between probability measures on the real line, is introduced. It generalizes Martingale optimal transport, and forms a hierarchy with the Hellinger and the Wasserstein metrics. We observe that certain classes of parabolic PDEs, among them the porous medium equation of exponent two, are formally semi-contractive metric gradient flows in the new distance. This observation is made rigorous for a suitable spatial discretization of the considered PDEs: these are semi-contractive gradient flows with respect to an adapted diffusive transport distance for measures on the point lattice. The main result is that the modulus of convexity is uniform with respect to the lattice spacing. Particularly for the quadratic porous medium equation, this is in contrast to what has been observed for discretizations of the Wasserstein gradient flow structure.