Let $X_K$ be a smooth proper curve over a field $K$, and let $S$ be a Dedekind scheme with function field $K$.
Let $X$ be the proper regular minimal model of $X_K$ over $S$.
Let $Y_K$ be another smooth proper curve over $K$, equipped with a finite map $Y_K\rightarrow X_K$.
Let $Y$ be the normalization of $X$ inside the function field of $Y_K$.
Must $Y$ be the proper regular minimal model of $Y_K$?
If not, what goes wrong, and are there reasonable situations where this is true? (perhaps if we assume $Y_K/X_K$ etale?)