For a random variable $X$ with cumulative distribution function $F$ one can show that $F(X)$ is uniformly distributed on $[0,1]$ by means of the quantile function. (see e.g. https://stats.stackexchange.com/questions/77845/inverse-function-for-a-non-decreasing-cdf for a quite nice explanation).
Now, what about $F(X+x)$ for $x\in \mathbb{R}$? What is its distribution? Any ideas?
Edit 1: Following the first comment, we obtain that $F(X+x)$ has cdf (denoted $\tilde F$) \begin{align} \tilde F (y) = P( F(X+x) \leq y) = P(X \leq F^{-1}(y) -x) = F(F^{-1}(y)-x ). \end{align} I don't see what distribution this is... But could we at least say something about the expectation of $F(X+x)$ then?