Smoothness of stabilisers in generic characteristic

Let $R$ be a commutative unital ring. Given a finitely-presented affine $R$-group $G$ acting on a finitely-presented $R$-scheme $X$ of finite type, we show that there is a prime $p_0$ so that for any $R$-algebra $k$ which is a field of characteristic $p > p_0$, the centralisers in $G_k$ of all subsets $U \subseteq X(k)$ are smooth. We prove this using the Lefschetz principle together with careful application of Gr\"{o}bner basis techniques.Comment: 15 page