$\DeclareMathOperator\Spec{Spec}$From studying the classical literature on deformation theory, I have the impression that if the identity morphism of a curve $C$ over an algebraically closed field $k$ has no non-trivial lift to the dual numbers $k[\epsilon]$, then for any relative curve $\pi: X \to S$, so $\pi$ being flat, proper, surjective, finitely presented morphism such that the fibre $X_s$ over any geometric point is a nodal projective, connected curve) over any base scheme with a geometric point $s: \Spec(k) \to S$ with $S$ connected s.t. $X_s \cong C$ there is no non-trivial automorphism which is the identity on $X_s$, however I cannot find a reference explicitly stating this result.

More general, is it true that a if two such morphisms between two such curves $f,g: X/S \to Y/S$ agree on all geometric points $x: \Spec(k) \to S$, then they are already the same morphism? I also have the impression that these questions are very closely linked, but I did not find a theorem about that.