To every Radon measure $\mu$  in $R^d$,  we associate a tangent bundle, which is a $\mu$-measurable multifunction $T_\mu$ from $R^d$  to vector subspaces of $R^d$. Inspired by Monge–Kantorovich optimal transport theory, this variational construction involves the duality between Lipschitz functions and the so called Arens-Eells space: a Banach subspace of distributions obtained by  completing  the set of balanced signed measures. We show that the tangent bundle $T_\mu$  is  local, which enables  the construction of a $\mu$-tangential gradient on Lipschitz funtions and a suitable  integration by parts formula. 
In this talk, after mentioning some applications in mechanics (multi-dimensional structures, composite materials, shape optimization), we demonstrate the fundamental nature of this construction  in light of its recent relation to the theory of $1$-Flat chains (the FCC conjecture in general metric spaces) and  the differentiability of Lipschitz functions a.e. with respect to $\mu$. Having retrieved the decomposability bundle recently introduced by Alberti and Marchese, we prove that $T_\mu$ characterizes the velocity fields along which the directional derivative of the Wasserstein distance vanishes.