Let $X, Y$ and $Z$ be measurable spaces. Let $P_1$ and $P_2$ be probability distributions on $X$ and $Y$ respectively and $P=P_1 \times P_2$. Let $h : X \times Y \to Z$ be a random variable. Let $A \subseteq X \times Y$ be measurable and let $A(x)$ be the x-section.
Is $P(A|h)(x,\cdot) = P_2(A(x)|h(x,\cdot))$ a.s. for $P_1$-a.a. x?
It suffices to show that $f: (x,y) \mapsto P_2(A(x)|h(x,\cdot))(y)$ is measurable, since then, by Fubini we have for $B \in \sigma(h)$ that $B(x)$ is $h(x,\cdot)$-measurable and
$$ \int_B f \, dP = \int \int_{B(x)} f(x,\cdot) \, dP_2 \, dP_1(x) \\ = \int \int_{B(x)} P_2(A(x)|h(x,\cdot)) \, dP_2 \, dP_1(x) \\ = \int \int_{B(x)} 1_{A(x)} \, dP_2 \, dP_1(x)\\ = \int_B 1_A \, dP $$
So $f=P(A|h)$ a.s.
We can easily show measurability if $Z$ is $[0,1]$ (or any polish space) by algebraic induction (characteristic function $\to$ simple function $\to$ measurable function)
So my question is for non-polish spaces.
I note that you could restate everything using sigma algebras and their section-sigma algebras.