  We study a dynamic optimal transport problem on a network. Despite the cost
for transport along the edges, an additional cost, scaled with a parameter
$\kappa$, has to be paid for interchanging mass between edges and vertices. We
show existence f minimisers using duality and discuss the relationship of the
model to other metrics such as Fisher-Rao and the classical Wasserstein metric.
Finally, we examine the limiting behaviour of the model in terms of the
parameter $\kappa$.

Analysis of PDEs