Motivation
Let $F$ and $K$ be distinct algebraically closed fields. Then $GL_n(F)$ acts on $M_n(F)$ by conjugation, and the nilpotent orbits of this action can be characterized by the Jordan normal form. Similarly for $GL_n(K)$ acting on $M_n(K)$, so that we have a bijection between the nilpotent orbits of $M_n(F)$ and $M_n(K)$ under their respective actions.
Background
Let $G$ be an affine group scheme over $\mathbb{Z}$, and let $X$ be an affine $\mathbb{Z}$-scheme on which $G$ acts via a map of schemes:
$$G\times X\to X$$
or equivalently via a map of rings
$$\varphi:\mathbb{Z}[X]\to \mathbb{Z}[G]\otimes\mathbb{Z}[X]$$
For a commutative ring $R$, we obtain an action of the $R$-points of $G$ on the $R$-points of $X$. Specifically, given $\rho:\mathbb{Z}[X]\to R$ and $\tau:\mathbb{Z}[G]\to R$, we can define $\tau\cdot\rho$ to be the $R$-point given by the composition
$$\mathbb{Z}[X]\xrightarrow{\varphi}\mathbb{Z}[G]\otimes\mathbb{Z}[X]\xrightarrow{\tau\otimes\rho}R\otimes R\to R$$
where the last map is multiplication in the ring. With this notion of an action, we can similarly define $R$-orbits (of $G$ on $X$).
Question
For two commutative rings $R$ and $S$, when can we expect there to be a bijection between the $R$-orbits and $S$-orbits? What are some reasonable hypotheses for $R$, $S$, $G$, $X$, and/or $\varphi$ to obtain such a bijection?
Edit: As stated in the comments, a bijection is a rather weak result to seek; there should be something richer. However, I don't know enough about group scheme actions to even hypothesize something else about two orbit spaces coming from the same action.
In general, I'd like to learn more about the similarities between different orbit spaces coming from the same group scheme action.