Given a infinite family of groups $(G_i)$ for $i\in I$. Is there a ring theoretic construction, that produces $R[\prod_{i\in I} G_i]$ using only the rings $(R[G_i])_{i\in I}$ ?

For the case of a finite family, we have $R[G\times H]\cong R[G][H]$ and for commutative $R$ we have $R[G\times H]\cong R[G]\otimes R[H]$. Neither of those constructions generalizes to the infinite case, e.g.

The map $R[\prod_i G_i]\rightarrow \mbox{invlim}_{I'\subset I, |I'|<\infty}R[\prod G_i]$ is not surjective (This product runs over $i\in I'$). The same holds for the map into the infinite tensor product (assuming that $R$ is commutative).

So I am hoping, that there is a better contruction in a more elaborate category (like $R$-Algebras with an augmentation), that produces $R[\prod_{i\in I} G_i]$ out of the group rings $(R[G_i])_{i\in I}$ .