Let $K$ be a number field. Suppose that $$ f: X \rightarrow Y $$ is a generically finite morphism of two-dimensional, normal schemes, which are projective and flat over $\operatorname{Spec} \mathcal{O}_K$. Assume further that the generic fibers $X_0,Y_0$ are smooth.

The morphism $$ f_0: X_0 \rightarrow Y_0 $$ is a finite morphism of smooth, projective curves, say of degree $n$. Furthermore, we can consider the base change $f_p: X_p \rightarrow Y_p$ for a prime $p \in \operatorname{Spec} \mathcal{O}_K \setminus \{0\}$. For all but finitely many primes $p$, this is also a finite morphism of smooth, projective curves. (right?)

Now to my question.

Is it true that for all but finitely many primes $p \in \operatorname{Spec} \mathcal{O}_K$, the morphism $f_p$ has degree $n$ as well?