We discuss the following nonholonomic version of the classical Moser theorem: given a bracket-generating distribution on a connected compact manifold (possibly with boundary), two volume forms of equal total volume can be isotoped by the flow of a vector field tangent to this distribution. We also present the Hamiltonian framework for the corresponding mass transport problem as an infinite-dimensional Hamiltonian reduction on diffeomorphism groups. The subriemannian heat equation turns out to be a gradient flow on the "nonholonomic" Wasserstein space with the potential given by the relative entropy functional. (This is a joint work with Paul Lee.)
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