Let $\mu$ and $\nu$ two probabilites on $\mathbb{R}^{d}$.

Let $T : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ a mesurable map such that $T_{\#} \mu = \nu$. I can disintegrate $\gamma := (id,T)_{\#} \mu$ according to $(h, h_{\#} \gamma$), I get a familly of measure $\gamma_{y}$ concentred on $h^{-1}(\{y\})$ as usually. But this time $\gamma$ have a special form. So do you think it can give me information about $\gamma_{y}$ ?

And with more generalities what do we get when we disintegrate a push forward measure $\gamma := f_{\#}\mu$ according $(h, h_{\#} \gamma)$ ?

Any help would be apprecieted, thanks and regards.

EDIT : As I have no answer, I'll try to be more specific, my question is, do we have $\gamma_{y} := (id,S)_{\#}\omega$ with $\omega$ a measure. I think $\omega$ could be a disintegration of $\mu$.