Given $N$ absolutely continuous probabilities $ρ_1, \dotsc, ρ_N$ over $\mathbb{R}^d$ which have Sobolev regularity, and given a transport plan $P$ with marginals $ρ_1, \dotsc, ρ_N$, 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.