Let K be the function field of a connected regular scheme S of dimension 1, and let $f : X\to Y$ be a finite cover of projective smooth and geometrically connected curves over K with $g(X)\ge 2$. Suppose that f can be extended to a finite cover ${\mathcal X} \to {\mathcal Y}$ of semi-stable models over S (it is known that this is always possible up to finite separable extension of K). Then there exists a unique minimal such cover. This gives a canonical way to extend $X\to Y$ to a finite cover of semi-stable models over S.
