Rlim versus tensor product

Let $R$ be a coherent ring, and let $(M_n)_{ngeq 1}$ and $(N_n)_{ngeq 1}$ be two inverse systems of finitely generated flat $R$-modules. If $R^1 lim M_n=R^1 lim N_n = 0$, is it true also that $R^1 lim (M_n otimes_R N_n)=0$? This seems very reasonable but I’m having trouble seeing it.

A more conceptual question: is there a reasonably general "Kunneth formula" for $Rlim$ on inverse systems of $R$-modules?