Let $f\colon X\to Y$ be a morphism of schemes and let $\mathcal{F}$ be an $\mathcal{O}_Y$-module. Let $s\in H^0(Y,\mathcal{F})$.
If I'm not mistaken, then $(f^{-1}s)_x=s_{f(x)}$ for all $x\in X$ and therefore $X_{f^{-1}s}=f^{-1}(Y_s)$.
Under which assumption can I make similar statements about the pullback section $f^*s$? Is it enough to require $X$ and $Y$ to be Noetherian and to restrict to the case $\mathcal{F}\in\text{Pic}(Y)$?
My motivation for the question is this answer on Mathoverflow which in my eyes seems to suggest that such a statement exists.