(I'm happy to work over an algebraically closed field....)

Let $\mathcal{C} \rightarrow Spec (R)$ be a (flat) family of proper, prestable curves where $R$ is a DVR. Suppose the generic fiber is smooth and the special fiber, $C_0$, is reduced but may be reducible.

Given a finite map of curves $f_0: D_0 \rightarrow C_0$ with $D_0$ also prestable, can this be extended to some map on some family?

That is, is there a flat family of proper curves $\mathcal{D} \rightarrow Spec(R)$ and an $R$-morphism $f: \mathcal{D} \rightarrow \mathcal{C}$ which reduces to $f_0$ on the special fiber?

Perhaps such an extension is possible only after a ramified cover of $Spec(R)$?

If so, can it be arranged that the generic fiber of $\mathcal{D}$ is smooth?