Tensor product of Koszul complexes

Question

I am trying to understand the tensor product of Koszul complexes. So, let $A$ and $B$ be $R$-modules over some commutative ring $R$. I am looking at the Koszul complex $K_{A\otimes_R B}(1 \otimes \gamma)$ where $\gamma$ is some endomorphism on $B$. I think it is true that this complex is quasi-isomorphic to the tensor product of complexes $K_A(1) \otimes_R K_B(\gamma)$. But is this true if we look at $K_{A\otimes_R B}(1 \otimes \gamma_1, \ldots, 1 \otimes \gamma_d)$ for some $d>1$? If it helps assumptions on $A$ and $B$ may be added...