  In this paper we study a dynamical optimal transport problem on a network
that allows for transport of mass between different edges if a penalty $\kappa$
is paid. We show existence of minimisers using duality and discuss the
relationships of the distance-functional to other metrics such as the
Fisher-Rao and the classical Wasserstein metric and analyse the resulting
distance functional in the limiting case $\kappa \rightarrow \infty$.
