Let $\mu$ and $\nu$ two probability measures on $\mathbb{R}^{d}$.
Let $\gamma \in \Pi(\mu,\nu)$, where $\Pi(\mu,\nu)$ is the subset of the probability measures $\pi$ such that $$ \pi (A\times Y) =\mu(a) \text{ and } \pi(X\times B)=\nu(B) $$ Disintegrating $\gamma$ according to $(h, h_{\#} \gamma$), I get a family of measures $\gamma_{y}$ concentrated on $h^{-1}(\{y\})$.
Can I say something about the marginals of $\gamma_{y}$?
I found something, not exactly what I need, maybe it could be a good start.
If $\gamma = \gamma_{y} \oplus \alpha_{\#} \gamma$ the disintegration of $\gamma$ according to $(\alpha, \alpha_{\#})$. Then $\beta_{\#}\gamma = \beta_{\#} \gamma_{y} \oplus \alpha_{\#} \gamma$
Now let $ \beta_{\#}\gamma = \mu_{y} \oplus \delta_{\#} \beta_{\#}\gamma$ the disintegration of $\beta_{\#} \gamma$ according to $(\delta, \delta_{\#} \beta_{\#} \gamma)$ then
$$ \mu_{y} = \beta_{\#} \gamma_{y} $$