Let $\mu, \nu$ be two probability measures on a space $X$ (assume Polish space). $T: X \rightarrow Y$ is a Lipschitz-map that acts as a push-forward on these measures; let $\mu^\prime = T_{\#\mu}$ and $\nu^\prime = T_{\#\nu}$ be the resulting push-forward measures on the space $Y$.

Let $d(\mu, \nu)$ denote some distance function between measures. What can be said about how $d(\mu, \nu)$ and $d(\mu^\prime, \nu^\prime)$ relate to each other? Especially, how is the relationship a function of the Lipschitz constant of $T$?

Of course the relationship is likely to be a function of the exact nature of $d()$ (e.g., KL-divergence, or Total-variation or Wassterstein .. ). I suspect a lot must be known about this type of questions; what would be the right place to look at?