Induced map in homology for a map to a loop space

Suppose $$Y$$ is an $$(n-1)$$-connected space, $$n>2$$, so we have Hurewicz isomorphisms $$\pi_n(Y)\cong H_n(Y)$$ and $$\pi_{n-1}(\Omega Y)\cong H_{n-1}(\Omega Y)$$. Let a map $$\alpha\colon X\to\Omega Y$$ be given. Naturally it induces a map $$\beta\colon X\times S^1\to Y$$. I want to show the following diagram is commutative: $$\require{AMScd} \begin{CD} H_{n-1}(X) @>\times[S^1]>> H_n(X\times S^1)\\ @V\alpha_*VV @V\beta_*VV \\ H_{n-1}(\Omega Y) @<\cong<< H_n(Y). \end{CD}$$

Since $$\Omega Y$$ is path connected, we may homotope $$\alpha$$ to a based map. Then $$\beta$$ factors though the reduced suspension $$\Sigma X$$. If $$X=S^{n-1}$$ is a sphere, the commutativity would then follow from tracking down the definition of $$\pi_n(Y)\xrightarrow{\cong}\pi_{n-1}(\Omega Y)$$. However I don't know how this helps for the general case.

One can also phrase the question in cohomology in the obvious way. (In particular the cross product $$\times[S^1]$$ will be replaced by the slant product $$/[S^1]$$.)

The general case does follow from the case $$Y=S^n$$.
Without loss of generality $$X=\Omega Y$$ and $$\alpha$$ is the identity. Now the statement comes down to the assertion that the composition $$\pi_{n-1}\Omega Y\to H_{n-1}\Omega Y\to H_n(\Omega Y\times S^1)\to H_n Y$$ corresponds to the Hurewicz map $$\pi_nY\to H_nY$$ via a natural isomorphism between $$\pi_{n-1}\Omega Y$$ and $$\pi_nY$$. This is true for all $$Y$$, and by naturality it is enough to prove it for $$Y=S^n$$.