Let $G$ be a connected reductive algebraic group and $X$ be a $G$-variety. Let $Y$ be a $G$-invariant irreducible sub variety of $X$ which has non-trivial intersection with the semi stable locus $X^{ss}$ (for a fixed line bundle $L$ on $X$). Let $Z$ be an irreducible codimension one sub variety of $Y$ which is not $G$-invariant and such that $Z$ lies strictly in $X\setminus X^{ss}$.

Question: What can be said about the $G$-sweep $G\cdot Z$, of $Z$. Under what conditions is it equal to $Y$?