Can we characterize a free action of a group scheme using $S$-points? Let $k$ be an algebraically closed field. All schemes in this post will be separated and finite type over $k$.
Let $G$ be a group scheme,  $X$ be a scheme, and suppose $G$ acts on $X$.
In Mumford's "Abelian Varieties" p. 103, he defined the action as free when
$$(\mu, p_2): G\times X\to X\times X$$
is a closed immersion. ($\mu:G\times X\to X$ is group action)
I want an equivalent condition considering $S$-valued points.
If $G\curvearrowright X$ is free, since closed immersion is monomorphism,
$$\underline{G}(S)\times \underline{X}(S)\to \underline{X}(S)\times \underline{X}(S)$$
is injectve. ($\underline{G}(S)={\rm Hom}(S,G)$).
This means $\underline{G}(S)\curvearrowright \underline{X}(S)$ is free as group action on set.
Is the converse true? That is, if for all schemes $S$, if $\underline{G}(S)\curvearrowright \underline{X}(S)$ is free, do we have $G\curvearrowright X$ free?
 A: Question: "I want an equivalent condition considering S-valued points."
Answer: Let $R$ be a fixed scheme. If $F,G: \underline{Sch/R} \rightarrow \underline{Sets}$ are functors and if $F \subseteq G$ is a"sub-functor" you define $F$ to be an"open subfunctor" (resp a "closed subfunctor") iff for any scheme $T \in \underline{Sch/R}$ and any morphism of functors $\rho: h_T \rightarrow G$ it follows the fiber product $h_T \times_G F \cong h_U$ where $U \subseteq T$ is an open subscheme (resp a closed subscheme).
Example: In your case you may define a morphism $\rho: h_{G\times_k X}\rightarrow h_{X\times_k X}$ and you may speak of $h_{G\times_k X}$ being a closed subfunctor.
Hence you define the action to be free iff the map $\rho$ is a closed immersion of functors. You may verify that if $F:=h_Y, G:=h_Z$, it follows $h_Y \subseteq h_Z$ is an open (resp closed) subfunctor iff $Y \subseteq Z$ is an open (resp closed) subscheme. This notion gives a general notion of "closed subfunctors" and you may use it to define free actions of group valued functors in greater generality - it gives a definition for
group valued functors that are not representable by a group scheme.
Note: You use this notion to define the notion "open cover $U_i$" of a functor $F$. One result is that a functor $F$ is representable (by a scheme) iff $F$ is a "sheaf in the Zariski topology" and $F$ has an "open cover" of representable  sub functors $h_{U_i}$. One application of this notion is to use it to prove that the relative projective space $\mathbb{P}(E^*)$ of a finite rank locally trivial sheaf $E$ exists and is a scheme. You define a functor $F_E$, prove it is a sheaf in the Zariski topology and use a local trivialization of $E$ to construct an "open cover" of $F_E$ by representable sub functors.
