  Let $k$ be a field, let $G$ be a reductive $k$-group and $V$ an affine
$k$-variety on which $G$ acts. In this note we continue our study of the notion
of cocharacter-closed $G(k)$-orbits in $V$. In earlier work we used a
rationality condition on the point stabilizer of a $G$-orbit to prove Galois
ascent/descent and Levi ascent/descent results concerning cocharacter-closure
for the corresponding $G(k)$-orbit in $V$. In the present paper we employ
building-theoretic techniques to derive analogous results.
