Let $K$ be a compact space, and let $A_i$ be a sequence of spaces in the following diagram. $$A_0 \hookrightarrow A_1 \hookrightarrow A_2 \hookrightarrow \cdots$$ All the inclusions in the diagram are closed embeddings. Does a continuous map from $K \to \mathrm{colim}\ A_i$ factor through some $A_j$ for $j \in \mathbb{N}$?
If it doesn't, can we strengthen the hypotheses so it does? In particular, I'm interested in knowing the answer for the special case where $A_i = \Omega^i \Sigma^i A_0$, where $\Omega$ is the loop space, and $\Sigma$ the suspension.