Suppose that $C_0$ is a dense open subvariety of a projective variety $C$, and that $C_0 \rightarrow D_0$ is a finite morphism. Does there exist a projective variety $D$ such that there is an extended morphism $C \rightarrow D$ that agrees with $C_0 \rightarrow D_0$ when restricted to $C_0$?
I understand that such a gluing can be done when $C_0$ is a closed subvariety of $C$. And that there are examples that show that this cannot be done in general. I am not looking for a counterexample as such, though these of course are welcome, but rather insight into how this might proceed in certain specific situations.