  This paper contains two contributions in the study of optimal transport on
metric graphs. Firstly, we prove a Benamou-Brenier formula for the Wasserstein
distance, which establishes the equivalence of static and dynamical optimal
transport. Secondly, in the spirit of Jordan-Kinderlehrer-Otto, we show that
McKean-Vlasov equations can be formulated as gradient flow of the free energy
in the Wasserstein space of probability measures. The proofs of these results
are based on careful regularisation arguments to circumvent some of the
difficulties arising in metric graphs, namely, branching of geodesics and the
failure of semi-convexity of entropy functionals in the Wasserstein space.

Analysis of PDEs;Metric Geometry;Probability