Continuous transformations of probability measures and their transport representations

Given a function F transforming a probability measure μ into another one F(μ), we study the existence and regularity of a transport representation of it. That is, we ask whether we can represent the image F(μ) of the input probability measure μ as the push-forward of μ by a map f(·,μ) which may depend on μ; and furthermore, how regular f can be chosen depending on F. Even if F is continuous and a transport representative exists, it cannot necessarily be chosen in a continuous way; however, if F is Lipschitz continuous with respect to the Wasserstein distance, then f can be chosen continuous. We provide several examples to illustrate the sharpness of our assumptions. This question is motivated by approximation results for transformations of probability distributions with transformers.