Let $X,Y$ be Polish spaces and suppose that $X$ is compact. Denote by $\mathcal{Mes}(X,\mathcal{P}(X\times Y))$ the set of (Borel) measurable functions from $X$ to the set of Borel probability measures $\mathcal{P}(X\times Y)$ on $X\times Y$. We equip this space with the uniform metric given on any two $f,g\in \mathcal{Mes}(X,\mathcal{P}(X\times Y))$ by $$ D(f,g) := \sup_{x \in X} \max\{1,d_{LP:Y}(f(x),g(x))\} , $$ where $d_{LP:Y}$ denotes the Lévy-Prokhorov metric on $\mathcal{P}(Y)$. Equip $\mathscr{P}(X)$ with the Lévy-Prokhorov metrics denoted by $d_{LP:X}$. Let $\pi:X\times Y \mapsto X$ be the natural projection $(x,y)\mapsto x$ and fix a measure $\mu \in \mathcal{P}(X\times Y)$.

By the Disintegration Theorem, we know that the disintegration of $\mu$ by $\pi$ associates to $\mu$ the ($\pi_{\#}\mu$-a.s.-unique) function $\operatorname{Dis}_{\pi}:x\mapsto \mu_x \in \mathcal{Mes}(X,\mathcal{P}(X\times Y))$. Therefore: $$ \operatorname{Dis}_{\pi}: \mathcal{P}(X\times Y)\mapsto \mathcal{Mes}(X,\mathcal{P}(X\times Y)). $$ Is this map ever continuous? I.e.: $$ \mbox{Does: } d_{LP:X}(\mu_n,\mu)\mapsto 0 \mbox{ imply that: } \sup_{x \in X} d_{LP:X\times Y}\left( \operatorname{Dis}_{\pi}(\mu_n)_x,\operatorname{Dis}_{\pi}(\mu)_x \right)\mapsto 0? $$

I expect this is false in general, but are there some references describing when it holds?