I believe you can get results along these lines if you replace Zar with something called the $\ell$dh topology which is a modification of the cdh topology. This topology depends on $\ell$ which is a prime that is not equal to the characteristic you are working over and is a replacement for resolution of singularities over finite fields. The result is originally in Shane Kelly's PhD thesis: http://arxiv.org/abs/1305.5349

And this is a consequence of corollary 4.25 of http://arxiv.org/pdf/1305.5690v2.pdf which states that every $M\mathbb{Z}_{(\ell)}$-module spectrum satisfies $\ell$dh descent.