Is the symmetric polynomial map
$$ \mathbb{A}^n/S_n \to \mathbb{A}^n $$
taking $\left(x_1,\ldots,x_n\right)$ to the list of elementary symmetric polynomials in the $x_i$ an isomorphism of fppf sheaves? Here the left hand side is the quotient of the fppf sheaf.
This is easily seen to be an isomorphism on points over a Henselian local ring with algebraically closed residue field if the left hand side is the presheaf quotient. In characteristic zero, it follows that this is an isomorphism on the (big) étale quotient. However, over an imperfect field it is easily seen to be nonsurjective.
I think it would help to have the following broader question. As I understand, fppf points are not well-understood. However, strictly Henselian local rings form "enough" pro-objects in that to check a map of fppf sheaves is an isomorphism, it suffices to check it is an isomorphism on all strictly Henselian local rings (though this is not a necessary condition). This class is not good enough to answer the above question; is there a more refined class of pro-objects which there are enough of in the above sense, does not contain this kind of inseparable object, and is useful in practice?