We are familiar with Künneth theorem:

The Kunneth formula is given by $R$ as a ring, $M,M'$ as the R-modules, $X,X'$ are some chain complex. The Kunneth formula shows the cohomology of a chain complex $X \times X'$ in terms of the cohomology of a chain complex $X$ and another chain complex $X'$. For topological cohomology $H^d$, we have $$ H^d(X\times X', M\otimes_R M') \simeq \Big[\oplus_{k=0}^d H^k(X,M)\otimes_R H^{d-k}(X',M')\Big]\oplus $$ $$\Big[\oplus_{k=0}^{d+1} \text{Tor}_1^R(H^k(X,M),H^{d-k+1}(X',M'))\Big] . $$

$$ H^d(X\times X',M) \simeq \Big[\oplus_{k=0}^d H^k(X,M)\otimes_{\mathbb Z} H^{d-k}(X',\mathbb{Z})\Big]\oplus $$ $$ \Big[\oplus_{k=0}^{d+1} \text{Tor}_1^{\mathbb Z}(H^k(X,{M}),H^{d-k+1}(X',\mathbb Z))\Big]. $$

The above is valid for both topological cohomology $H^d$ and group cohomology $\mathcal{H}^d$ (for $G'$ is a finite group): $$ \mathcal{H}^d(G\times G',M) \simeq \Big[\oplus_{k=0}^d \mathcal{H}^k(G,M)\otimes_{\mathbb Z} \mathcal{H}^{d-k}(G',\mathbb Z)\Big]\oplus $$ $$ \Big[\oplus_{k=0}^{d+1} \text{Tor}_1^{\mathbb Z}(\mathcal H^k(G,M),\mathcal H^{d-k+1}(G',\mathbb Z))\Big]. $$

Questions:[1]. Do we have similar results of Künneth theorem for bordism groups $\Omega_d^{...}(...)$? Schematically, maybe something like $$ \Omega_d^{...}(...) \simeq \oplus_n \Omega_n^{...(1)}(...) \otimes \Omega_{d-n}^{...(2)}(...)? $$

[2]. Can we, and, how can we interpret the decompositions of bordism group generators as manifolds $\Sigma_d$: $$ \Sigma_d \sim \Sigma_{d-n}^{(1)} \times \Sigma_{n}^{(2)}? $$ where $d$-manifold generators are related to the $d-n$-manifold generator and $n$-manifold generator. Are these general, or are they only special cases?

Refs are welcome. Thanks.