Suppose that $X$ and $Y$ are compact metric spaces. A Borel probability measure $\mu$ on $X\times Y$ satisfies $$ \mu(A\times B)=\int_A\mu(B|x)\mu_X(dx), $$ for $A$ and $B$ Borel sets in $X$ and $Y$ respectively, where $\mu(\cdot|x)$ is a transition probability measure, i.e., a measurable map $x\mapsto\mu(\cdot|x)$ from $X$ to the set of Borel probability measures on $Y$ (with the topology of weak convergence of measures) and where $\mu_X$ is the marginal of $\mu$ on $X$. The transition measure $\mu(\cdot|x)$ is continuous if the map $x\mapsto\mu(\cdot|x)$ is continuous. Note that here the map $x\mapsto\mu(\cdot|x)$ is a map between compact metric spaces.
Now suppose that $X=X_1\times X_2$ and $Y=Y_1\times Y_2$ ($X_1$, $X_2$, $Y_1$, $Y_2$ compact and metric). A transition probability measure $\mu(\cdot|x)$ is a product transition probability measure if $$\mu(B_1\times B_2|(x_1,x_2))=\mu_1(B_1|x_1)\mu_2(B_2|x_2)$$ for $B_1$ and $B_2$ Borel subsets of $Y_1$ and $Y_2$ respectively, where $\mu_1(\cdot|x_1)$ is a transition probability measure with source $X_1$ (with its Borel sets) and target $Y_1$ (with its Borel sets) and $\mu_2(\cdot|x_2)$ is a transition probability measure with source $X_2$ (with its Borel sets) and target $Y_2$ (with its Borel sets).
Endow the set $\mathfrak C$ of continuous transition probability measures with the topology of uniform convergence (which renders the space metric, complete and separable), and let $\mathfrak C^*$ be the subspace of continuous product transition probability measures, which is also complete and separable.
Let $\{\mu_n(\cdot|x)\}$ be an arbitrary sequence of continuous product transition probability measures and define for each $n$ a Borel probability measure $\nu_n$ on $\mathfrak C^*$ by $$ \nu_n:=\frac{1}{n}\delta_{\mu_1}+\cdots+\frac{1}{n}\delta_{\mu_n}, $$ where $\delta_{\mu_k}$ denotes the Dirac measure with pointmass on the transition measure $\mu_k$. The measure $\nu_n$ induces a (continuous) transition probability measure $\nu_n(\cdot|x)$ defined by $$\nu_n(B|x):=\frac{1}{n}\mu_1(B|x)+\cdots+\frac{1}{n}\mu_n(B|x).$$ Suppose that this transition measure converges uniformly to a (continuous) transition measure $\nu(\cdot|x)$. Is there a Borel probability measure $\lambda$ on $\mathfrak C^*$ such that for each $x\in X$ and $B$ a Borel set in $Y$, $$ \nu(B|x)=\int_{\mathfrak C^*}\mu(B|x)\lambda(d\mu)? $$
