There are several equivalent definitions of Sobolev functions valued in a metric space X, e.g. given by N. Korevaar and M. Schoen in 1993 or by Y. Reshetnyak in 1997. In 2019 H. Lavenant proved a multidimensional analogue of the Benamou-Brenier formula in case where X is the Wasserstein space on R^d. In this talk we consider a notion of convergence of Sobolev measure-valued functions related to this result. We also discuss some fine properties of these functions, such as stability of precise representatives, and its application to an optimal transport problem penalized with (a relaxation of) the Dirichlet energy of a transport map.