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.

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, also see Schmidt - Vershik or Neretin).