Let $(X_n,Y_n)$ be a sequence of random vectors in $\mathbb R^k\times \mathbb R^l$ such that $X_n\to X$ in distribution. Denote the conditional distributions $Y_n$ given that $X_n=x$ by $P_{n,x}$. I want to find a condition on $P_{n,x}$ that would guarantee joint convergence in distribution of $(X_n,Y_n)$ in $\mathbb R^{k+l}$.
The condition I came up with is the following:
(a) $P_{n,x}=P_x$ does not depend on $x$; and
(b) There exists a set $A\subseteq\mathbb R^k$ with $P(X\in A)=1$ such that for all $x\in A$ and all $x_n\to x$, $P_{x_n}\to P_x$ strongly, i.e., $P_{x_n}(B)\to P_x(B)$ for any Borel set $B\subseteq \mathbb R^l$. This implicitly assumes that $P_{y}$ is defined for $y$ sufficiently close to $x$.
I could show that (a)+(b) yield joint convergence of $(X_n,Y_n)$. My question is to what extent condition (b) is necessary. Clearly (a) is far from necessary but I do not mind relaxing it, I am really interested in (b). In particular, would it work if I replaced strong convergence by convergence in distribution? I think not, but could not come up with a counter-example.
Some regularity conditions. I am okay with imposing the extra assumptions that $Y_n$ are uniformly bounded (so the sequence $(X_n,Y_n)$ is automatically tight) and that $X$ (but not $X_n$ or $Y_n$) is absolutely continuous with positive density througout.
I know very little about strong convergence: it is weaker than total variation and stronger than convergence in distribution, it is not metrizable. But I feel like a better understanding of strong convergence can help with my problem, so also want to learn more about it. Is there a reference (preferably a textbook) that deals with it systematically? What is an example of a sequence that converges strongly but not in total variation? (The empirical measure from a continuous distribution function is the "canonical" example of convergence in distribution that does not hold strongly).
An alternative route is to understand what continuity properties of the mapping $x\mapsto P_x$ can be obtained from the disintegration theorem?