We study a variant of the dynamical optimal transport problem in which the energy to be minimized is modulated by the covariance matrix of the current distribution. Such transport metrics arise naturally in mean field limits of recent particle methods for inverse problems. We show that the transport problem splits into two separate minimisation problems: one for the evolution of mean and covariance of the interpolating curve and one for its shape. The latter consists in minimizing the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyze the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. This is joint work with Martin Burger, Matthias Erbar, Daniel Matthes and André Schlichting.