If $X_1\supseteq X_2\supseteq \ldots$ is a sequence of "nice" compact spaces, I would like to know whether the natural map from $H_*(\cap X_i)$ to the inverse limit $\lim \, H_*(X_i)$ is surjective. In particular, if there exist nonzero $\beta_i\in H_q(X_i)$ such that the inclusion-induced homomorphism $i_*$ takes $\beta_i$ to $\beta_{i-1}$, is there an $\alpha\in H_q(\cap X_i)$ such that the inclusion-induced image of $\alpha$ is the $\beta_i$ for each $i$?

I have found Milnors paper "On the Steenrod homology theory" where he shows, using Steenrod homology theory, that there exists a surjective map $H_q(\cap X_i)\to \lim H_q(X_i)$. However, it is not completely clear from the text whether the preimage of $(\beta_1,\beta_2,\ldots)\in \lim H_q$ is mapped to $\beta_i$ by the inclusion-induced homomorphism $H_q(\cap X)\to H_q(X_i)$.