  We introduce and study a new optimal transport problem on a bounded domain
$\bar\Omega \subset \mathbb R^d$, defined via a dynamical Benamou-Brenier
formulation. The model handles differently the motion in the interior and on
the boundary, and penalizes the transfer of mass between the two. The resulting
distance interpolates between classical optimal transport on $\bar\Omega$ on
the one hand, and on the other hand between two independent optimal transport
problems set on $\Omega$ and $\partial \Omega$.
