Let $K$ be an algebraically closed field of characteristic $0$, let $X/K$ and $Y/K$ be quasi-projective varieties, and let $f:X\to Y$ be a morphism. Let $G/K$ be a reductive group that acts stably on $X$ and $Y$, i.e., every point of of $X$ and $Y$ is in the $G$-stable loci in the sense of GIT. Assume further that the action of $G$ commutes with the map $f$. We then get an induced map on the GIT quotient varieties $$ \overline{f} : X//G \longrightarrow Y//G. $$ (More precisely, $X$ and $Y$ come with line bundles $\mathcal{L}_X$ and $\mathcal{L}_Y$ that are used to linearize the $G$-actions when forming the quotients, and we'll assume that $f$ respects those linearizations, too.)
Question 1: If $f:X\to Y$ is a finite map, is it always true that the induced map $\overline{f} : X//G \to Y//G$ is a finite map?
Question 2: If not, what additional conditions on $X$, $Y$, $f$ and/or the action of $G$ would ensure that $\overline{f} : X//G \to Y//G$ is finite?
If the answer to Question 1 is in the affirmative, a reference would be much appreciated; or failing that, a proof sketch.
