Étale geometry of closures of Jordan classes

Let $G$ be a connected reductive algebraic group with simply connected derived subgroup. Over the complex numbers there exists a local method to study the geometric properties of a point $g$ in the closure of a Jordan class of $G$ in terms of Jordan classes of a maximal rank reductive subgroup $M\le G$ depending on the point $g$, and further to the closures of certain decomposition classes in Lie($M$). We adapt this method to the case of an algebraically closed field of characteristic $p$, and we give sufficient restrictions on $p$ for it to hold.