A coproduct obstruction for derived unramified cohomology

Let $k$ be a perfect field of exponential characteristic $p$, and let $R$ be a commutative $\mathbb{Z}[1\mathrm{/}p]$-algebra. We prove that the first derived unramified functor $$F⟼R_{\mathrm{n}\mathrm{r},R}^{1}F$$ from homotopy invariant Nisnevich sheaves with transfers of $R$-modules to birational sheaves commutes with arbitrary small direct sums. This gives a positive answer, after inverting the exponential characteristic, to a question of Kahn and Sujatha; on smooth projective varieties no inversion is needed. We also describe an obstruction to this for the functor $R_{\mathrm{n}\mathrm{r},R}^2$ in categorical terms, which includes the familiar Griffiths group obstruction. As applications of the motivic nature of the functors $R_{\mathrm{n}\mathrm{r}}^q$, we prove torsion-order bounds and a correspondence-detection statement for surfaces.