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*?