Let $E \to \operatorname{Spec} K$ be an elliptic curve and $S$ be its Neron model. That is, a scheme $S$ over a Dedekind domain $R$ with $E/K$ as the generic fiber of the structure morphism $N \to \operatorname{Spec}R$. Given the addition morphism $n_E:E\to E$, how would one go about showing that this can be extended to a morphism on a dense open subset of $S$?
This fact is used in Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, Theorem IV.5.c, but it is stated without proof. I have (so far) been unable to prove it myself or find a reference, although it seems frustratingly simple.
This is really just a long comment and in no way a complete answer. I hope this is slightly helpful.
Let $X, Y$ be irreducible schemes over $Spec(R)$, where $R$ is a domain. Let $K = Q(R)$. Let $X', Y'$ be the generic fibers. Let $f : X' \rightarrow Y'$ be a morphism of schemes over $Spec(K)$. Then
There exists a multiplicative set $S \subset R$ and a morphism $g : X \times_{R} S^{-1}R \rightarrow Y \times_{R} S^{-1}R$, such that $g \times Spec(K) = f$.
This statement certainly implies the statement required in the question. Hence we are reduced to proving the statement above.
Step 1. Reduce to the case when $Y$ is affine.
Step 2. Reduce to the case of $X$ is affine.
Step 3. Given a map $Spec(B \otimes_R K) \rightarrow Spec(A \otimes_R K)$, where $A, B$ are finitely generated $R-$algebras, then there exists a multiplicative set $S \subset R$, and a homomorphism $g : S^{-1}A \rightarrow S^{-1}B$ such that $g \otimes K = f$.