Let $X$ and $T$ be subset of $\mathbb{R}^d$ and $\mathbb{R}^k$, respectively, where $d,k$ are positive integers. Endow them with their Borel $\sigma$ algebras, $\mathcal{B}_X$ and $\mathcal{B}_T$, respectively. Let $p(x;t)$ be the Lebesgue probability density associated to a Markov kernel $P(\cdot;\cdot)$ on $X\times T$: i.e.
for every fixed $B\in \mathcal{B}_X$, the map $t \mapsto P(B;t)$ is $\mathcal{B}_T$-measurable;
for every fixed $t \in T$, the map $B \mapsto P(B;t) $ is a probability measure on $(X,\mathcal{B}_X) $.
Thus, we could think of $p(x;t)$ as a conditional probability density.
Now, consider the quotient topological space $T^*$ where each two $t\in T$ and $t' \in T$ such that $$ p(x;t)=p(x;t'), \quad \text{for almost every } x, $$ are seen as the same element. Let $Q$ be a probability measure on $(T, \mathcal{B}_T)$, with Lebesgue density $q$, and let $\phi:T \mapsto T^*$ be the map associating to each $t\in T$ the corresponding representative in $T^*$; such a map may be not $1-\text{to}-1$.
QUESTION How does one obtain the probability density of $Q \circ \phi^{-1}$? Under which conditions does such density exist? If it exists, how does it relate to $q$?