Abstract: Week topology implicit schemes based on Monge-Kantorovich or Wasserstien metrics have become prominent for their ability to solve a variety of diffusion and diffusion-like equations. They are very flexible, encompassing a wide range of nonlinear effects. They have interesting interpretations as descent algorithms in an infinite dimensional manifold setting or as dissipation principles for motion in a highly viscous environment. Transport plays a fundamental role in these schemes, as noted by Brenier and Benamou and reviewed below. The reverse implication is less explored and, at least at the outset, less obvious. Here we discuss the simplest situations and provide a few applications.