A categorical Künneth formula for constructible Weil sheaves

We prove a Künneth-type equivalence of derived categories of lisse and constructible Weil sheaves on schemes in characteristic $p>0$ for various coefficients, including finite discrete rings, algebraic field extensions $E\supset {\mathbb{Q}}_{\ell }$, $\ell \ne p$, and their rings of integers ${\mathcal{𝒪}}_E$. We also consider a variant for ind-constructible sheaves which applies to the cohomology of moduli stacks of shtukas over global function fields.