I get that the answer is "no" for an abelian variety over the algebraic closure of F_p with complex multiplication by a ring of rank greater than 2. Since you say you have already thought through the abelian variety case, I wonder whether I am missing something.

More generally, let X be any variety over the algebraic closure of F_p with an automorphism f of infinite order. A concrete example is to take X an abelian variety with CM by a number ring that contains units other than roots of unity. Any finite collection of closed points of X will lie in X(F_q) for some q=p^n. Since X(F_q) is finite, some power of f will act trivially on X(F_q). Thus, any finite set of closed points is fixed by some power of f.

As I understand the applications to descent theory, this is still uninteresting. For that purpose, we really only need to kill all automorphisms of finite order, right?

I get that the answer is "no" for an abelian variety over the algebraic closure of F_p with complex multiplication by a ring with a unit of infinite order. Since you say you have already thought through the abelian variety case, I wonder whether I am missing something.

More generally, let X be any variety over the algebraic closure of F_p with an automorphism f of infinite order. A concrete example is to take X an abelian variety with CM by a number ring that contains units other than roots of unity. Any finite collection of closed points of X will lie in X(F_q) for some q=p^n. Since X(F_q) is finite, some power of f will act trivially on X(F_q). Thus, any finite set of closed points is fixed by some power of f.

As I understand the applications to descent theory, this is still uninteresting. For that purpose, we really only need to kill all automorphisms of finite order, right?