We all are aware of the change of variable formula whereby if $$[A, B] = g(X, Y) $$ and $g$ are invertible, then the joint density function of $A$, $B$ is given by $$f_{ab} (A, B) =\frac1{|J|} f_{XY} (g^{-1}(a, b) ),$$ where $J$ is the Jacobian.
However, if the joint distribution of $X, Y$ is not known but only the conditional distribution of $X$ given $Y$ is known, can we find the joint distribution of $A, B$ given $Y$.