Let $G_i$ be a compact Lie group, $i=1,2$, and let $E_{G_i}^*$ be a $\mathbb{Z}$-graded complex-oriented $G_i$-equivariant generalized cohomology theory with commutative products. Let $X_i$ be a compact $G_i$-manifold. Note that $X_1\times X_2$ is then a $(G_1\times G_2)$-manifold, and we may consider $E_{G_1\times G_2}^*(X_1\times X_2)$.

$\textbf{Question}:$ In what sort of generality can we say that $E_{G_1\times G_2}^*(X_1\times X_2)$ and $E_{G_1}^*(X_1)\otimes_{\mathbb{Z}} E_{G_2}^*(X_2)$ are naturally isomorphic as rings? (This question is vague, so please feel free to answer in any way you desire.) In particular, is this true if $G_1$ and $G_2$ are tori and the theory in question is equivariant complex cobordism? I would also be grateful for a reference.

This is not difficult to verify in the case of equivariant cohomology with rational coefficients. It also holds in equivariant K-theory.