Since you refer to transportation metrics, it seems to be fair to assume that your probability spaces are Lebesgue (=standard). Then it is the same to talk either about "lifting" your measure $\mu$ to $\gamma$ or about the **family of the conditional measures** of the projection $\gamma\to\mu$ (or,  which is also the same, about the corresponding **Markov kernel**). Ordinary functions from $X$ to $Y$ correspond then to the situation when all transition measures are delta-measures.

This construction is known as **mutivalued maps with invariant measure** or **polymorphisms** (the term introduced by [Vershik][1], also see [Schmidt - Vershik][2] or [Neretin][3]).


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=2166671
  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=2408396
  [3]: https://mathscinet.ams.org/mathscinet-getitem?mr=1944085