The inner structure of realistic materials make them exhibit momentum relaxation. In this paper we study the Joule-Thomson effect on AdS black holes in which translational invariance is broken by two methods: First by considering planar black holes in general relativity supported by scalar fields with a linear dependence on the horizon coordinates and secondly by considering black holes in massive gravity models in which momentum relaxation is obtained by breaking the bulk diffeomorphism invariance of the theory. In contrast with black holes studied so far, for both theories it is possible to obtain inversion curves with two branches reproducing the behavior of Van der Waals fluids. Moreover in the specific case of the massive gravity model we show that black holes can heat up when crossing the inversion curve.