Orbits that 'coalesce' Let $R$ be a commutative ring, $G$ a group scheme over $\mathrm{Spec}\;R$, and $X$ a scheme over $\mathrm{Spec}\;R$ on which $G$ acts $R$-morphically via $G\times X\to X$.  Suppose $S$ is another commutative ring together with a fixed ring map $R\to S$.  The commutative diagram
$$\begin{array}{ccc}
G(R)\times X(R)&\longrightarrow&X(R)\\\downarrow&&\downarrow\\G(S)\times X(S)&\longrightarrow&X(S)\end{array}$$
shows that if $x,y\in X(R)$ are in the same $G(R)$-orbit, then $x_S,y_S\in X(S)$ are in the same $G(S)$-orbit.  Here $x_S$ denotes the image of $x$ under the map $X(R)\to X(S)$ obtained from functoriality of $X$.  The converse however is not true, because it is possible for two or more $G(R)$-orbits to 'coalesce' into one $G(S)$-orbit.  For example, if $G=\mathrm{GL}_2$, $X$ is the scheme of $2\times 2$ nilpotent matrices on which $G$ acts by conjugation, $R=\mathbb{Z}$, and $S=\mathbb{C}$, there are infinitely many $G(R)$-orbits, but only $2$ $G(S)$-orbits.
What are some sufficient conditions that lead to two orbits 'upstairs coalescing downstairs?'  Has this been studied before?  Does anybody have a reference, or a few basic results towards answering my general question?
 A: This is a huge topic.  

One example that comes to mind is when $R = F,$ and $S = \overline{F}$, its algebraic closure, and $X = G$, with the $G$-action being conjugation.
One then says that two elements of $G(F)$ are stably conjugate if
they become conjugate in $G(\overline{F})$.
If $G = \mathrm{GL}_n$, then it's not too hard to check that conjugacy
and stable conjugacy are the same, but for basically any other reductive
group they are different, e.g. already this is the case for $\mathrm{SL}_n$.
The distinction between conjugacy and stable conjugacy (the "coalescing" you ask about) can often be measured
by certain Galois cohomology groups.  The places to look are the original paper of Langlands--Labesse on $\mathrm{SL}_2$, and maybe papers of Kottwitz for more general contexts.  
(This theory arises as part of the general theory of "stablizing the trace formula" in the Langlands program, and there is a huge amount of work on it;
it may not be that accessible to an outsider, though.)

The passage from $F$ to $\overline{F},$ when $X$ is a representation of $G$,comes up in the work of Bhargava, Gross, and their collaborators.  These papers are probably representative.  You can see that Galois cohomology again plays an important role.

Another example comes from taking $R = \mathbb Q$ and $S = $ one of its completions ($\mathbb R$ or one of the fields $\mathbb Q_p$).  Then one
is asking when different global orbits coincide locally.  This comes up
a lot when $X$ is the space of quadratic forms in $n$ variables and $G = \mathrm{GL}_n$.
A related example occurs when $R = \mathbb Z$ and $S = \mathbb Q$, where we again take $X$ to be the space of quadratic forms in $n$ variables, and $G$ to be $\mathrm{GL}_n$ or $\mathrm{SL}_n$.
Then this is related to the theory of genera of quadratic forms.  (Two quadratic forms over $\mathbb Z$ are in the same genus if they are equivalent over $\mathbb R$ and also over $\mathbb Z_p$ for all primes $p$; note that by Hasse--Minkowski this implies in particular that they are equivalent over $\mathbb Q$.)
