This question is broadly about non-trivial examples where a map $f:Y\to X$ of smooth projective $k$-varieties ($k$ not algebraically closed) is such that existence of $k$-points (or in fact $k'$-points for $k'/k$ a finite extension) on $X$ ensures their existence on $Y$. For example, if $f$ is a birational $k$-map, the Lang-Nishimura Theorem guarantees an element of $Y(k)$ if $X(k)$ is not empty.

An interesting example is the case of $f:Y\to X$ being a finite étale cover. In general it will not be the case that a point in $X(k)$ lifts to one in $Y(k)$ (for example elliptic curves and an element in the Tate-Shafarevich group). Are there however examples or conditions that guarantee this happens?

We say that a variety $X/k$ is potentially dense if $X(k')$ is dense in $\bar{X}$ for some finite extension $k'/k$. Campana's classification program conjectures that $X$ is potentially dense if and only $X$ is special (i.e. has trivial core), so in this case, since it seems that a finite étale cover remains special if the base is special, we'd have that the arithmetic of special varieties is preserved under étale covers. But does potential density pull back under finite étale covers?

Related bibliography and any comments on the aforementioned or related topics mentioned are welcome!