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?)