Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic 0 (a reduced separated scheme of finite type over $k$). Let $G$ be a connected linear algebraic group over $k$ (an connected affine group scheme of finite type over $k$) acting on $X$.
I am looking for a proof or a reference to a proof of the following lemma:
Lemma. There exists a stratification $$ X=\bigsqcup_{i\in I} X_i $$ of $X$ into a finite union of $G$-invariant non-intersecting locally closed subvarieties $X_i$ with the following properties:
(1) Each $X_i$ is irreducible and smooth.
(2) For each $i$ there exists a surjective morphism $f_i\colon X_i\to Y_i$ onto a smooth $k$-variety $Y_i$ whose fibres are orbits of $G$ in $X_i$.
(3) Each morphism $f_i$ is flat, or, equivalently, all orbits of $G$ in $X_i$ have the same dimension $n_i$ depending only on $i$.
EDIT. What else can be required for a stratification to be good?
