CDF of a random variable evaluated at a differently distributed random variable Let $X$ be a random variable on $\mathbb{R}$ and $Y$ another random variable, independent from $X$ and also defined on $\mathbb{R}$. Let $F_X$ be the CDF of $X$. I'm interested in calculating the distribution of $F_X(Y) = \mathbb{P}(X \leq Y \mid Y)$. Is there a general way of calculating this?
We know that if $X$ and $Y$ are continuous and identically distributed, $F_X(Y) \sim Uniform[0, 1]$, but what about in general? Or maybe, if we look at a simplified example:
Let $X \sim N(0,1)$, $Y \sim N(\mu, 1)$, $X$ independent of $Y$, what is the distribution of $F_X(Y)$? 
On this particular example, I did some simulations in R programming language and I have a sense that $F_X(Y)$ is Beta distributed, but couldn't find a way to prove it. I also obtained similar results when $X$,$Y$ were Chi-squared, Exponential, or Weibull distributed.
Any help or further literature is greatly appreciated!
 A: The cdf of $Z=F_X(Y)$ is
\begin{align}
F_Z(z)
  &=P(Z\le z)
\\&=P(F_X(Y) \le z)
\\&=P(Y \le F_X^{-1}(z))
\\&=F_Y(F_X^{-1}(z)).
\end{align}
If $F_X$ is not continuous and strictly increasing, for example if $X$ is discrete, then $F_X^{-1}$ is the generalised inverse of $F_X$, that is, $F_X^{-1}(z)=\mathrm{inf}(y \in \mathbb R:F_X(y)\ge z)$, also known as the quantile function of $X$.
In the case that $X$ and $Y$ are both continuous and the support of $Y$ is a subset of the support of $X$, the pdf of $Z$ becomes
$$
f_Z(z) = \frac d{dz}F_Z(z)=f_Y(F_X^{-1}(z))\frac d{dz}F_X^{-1}(z)=\frac{f_Y(F_X^{-1}(z))}{f_X(F_X^{-1}(z))}. \tag{1}
$$
If both $X$ and $Y$ are continuous but $Y$ has larger support than $X$, for example if $X\sim\mathrm{exp}(1)$ and $Y\sim N(1,1)$, then $Z$ has a pdf given by (1) but in addition a point mass of size $P(Y\le 0)=\phi(-1)=0.1586$ in $z=0$.
R code illustrating this:
n <- 1e+4
X <- rexp(n,shape=1,scale=1)
Y <- rnorm(n,mean=1)
Z <- pexp(Y)
hist(Z,prob=TRUE)
curve(dnorm(qexp(z),mean=1)/dexp(qexp(z)),xname="z",add=TRUE)


