If X is a smooth separated scheme over R then any K-morphism from XK to AK can be extended to a unique R-morphism from X to AR (Néron mapping property).
In terms of sheaves, any scheme A over Spec(K) represents a sheaf on the category of schemes smooth over Spec(K) with the smooth Grothendieck topology, and this has a pushforward by the injection map from Spec(K) to Spec(R), which is a sheaf over Spec(R).