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The only reference I know for Künneth theorems in this generality are SGA4 and SGA4.5. Specifically, SGA4 has theorems for étale cohomology with proper support (which hold in ridiculous generality), but SGA4.5's "Théorèmes de finitude en cohomologie $l$-adique" allows you to get the same results for étale cohomology, provided that the schemes are of finite type over a field (no other assumptions are needed!). The basic Künneth theorem itself is given as Corollaire 1.11 in that exposé. It is stated for schemes of finite type over a separably closed field, but if I'm not mistaken the proof works over any field whatsoever if you replace global sections by pushforwards to the étale topos of the base field. So this only gives you an equivalence between chain complexes with an action of the Galois group. The actual "étale cochains" are the invariants of those, and I don't know how to compute the invariants of a tensor product. But at least this reduces the problem to one in group cohomology.