We write down a parabolic optimal transport equation and prove that, in almost all of the cases where regularity is known in the elliptic case, the solutions exists for all time and converge to a solution of the elliptic optimal transport equation. Using a metric motivated by special Lagrangian geometry, exponential convergence follows quite easily from an argument of Li-Yau. We will discuss this result, as well as some motivations and analogies to special Lagrangian geometry. We will focus on joint work with Young-Heon Kim and Jeffrey Streets, and may also mention work with Kim and Robert McCann.