  Given $N$ absolutely continuous probabilities $\rho_1, \dotsc, \rho_N$ over
$\mathbb{R}^d$ which have Sobolev regularity, and given a transport plan $P$
with marginals $\rho_1, \dotsc, \rho_N$, we provide a universal technique to
approximate $P$ with Sobolev regular transport plans with the same marginals.
Moreover, we prove a sharp control of the energy and some continuity property
of the approximating family.
