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.
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