  We modify the JKO scheme, which is a time discretization of Wasserstein
gradient flows, by replacing the Wasserstein distance with more general
transport costs on manifolds. We show when the cost function has a mixed
Hessian which defines a Riemannian metric, our modified JKO scheme converges
under suitable conditions to the corresponding Riemannian Fokker--Planck
equation. Thus on a Riemannian manifold one may replace the (squared)
Riemannian distance with any cost function which induces the metric. Of
interest is when the Riemannian distance is computationally intractable, but a
suitable cost has a simple analytic expression. We consider the Fokker--Planck
equation on compact submanifolds with the Neumann boundary condition and on
complete Riemannian manifolds with a finite drift condition. As an application
we consider Hessian manifolds, taking as a cost the Bregman divergence.
