Let $K/k$ be a field extension, let $V_0$ be a variety over $k$, and let $V=V_0\times_k\mathrm{Spec}\;K$, so that we can speak of the $k$-rational points of $V$ as morphisms $\mathrm{Spec }\;k\to V_0$. Suppose $G$ is an algebraic group (also defined over $k$) acting morphically on $V$, which partitions $V$ into orbits:
$$V=\bigsqcup_{i\in I}\mathcal{O}_i$$
It can be shown that the orbits themselves are all smooth $G$-varieties over $K$, which also have a $k$-structure, i.e., there are $G$-varieties over $k$, $\mathcal{O}_{i,0}$, such that $\mathcal{O}_i\cong\mathcal{O}_{i,0}\times_k\mathrm{Spec}\;K$. In this way, we can speak of the $k$-rational points of $\mathcal{O}_i$ to be morphisms $\mathrm{Spec}\;k\to\mathcal{O}_{i,0}$.
If I can write down all of the $k$-rational points of $V$, and I know which of them belong to which $\mathcal{O}_i$, can I say anything about the above partition? Specifically, I'm interested in computing the dimensions of the $\mathcal{O}_i$, and I'd like to know when $\mathcal{O}_i$ is in the closure of $\mathcal{O}_j$.
Perhaps there is not enough data in the $k$-rational points to conclude anything about the orbits. If so, what other information would I need to compute dimensions and closures?