Let $C$ and $D$ be two algebraic curves defined over a function field $K(t)$ ($K$ is a number field). Suppose that $C$ and $D$ are isomorphic. We can specialize the curves to values in $K$ (nonformally this can be done by just pluggin in a number in $K$ instead of $t$). This scenario is the same as reducing from a generic fiber to a special fiber. Is it true that the specialization of the isomorphism will remain an isomorphism for all but finitely many values in $K$?
I am not sure if this is a natural thing to do, or if this is the best way to present the question. I appreciate any help on this.