Let $B_n$ be the braid group on $n$ strands, $B_{\infty}$ the direct limit of braid groups. For a discrete group $G$, we let $BG$ to be the classifying space of $G$.

After reading this question, I was curious about the following :

Why is the group completion of $\coprod_{k \geq 0} BB_{k}$ homotopy equivalent to $\mathbb{Z} \times BB_{\infty}^{+}$?

(I think I've seen such things come up fairly often in algebraic topology - where you have symmetric groups $\Sigma_n$, $\operatorname{GL}_n(R)$, mapping class groups, etc instead of braid groups here. Can we somehow generalize all of them?)

PS) Let me edit the last part. After skimming through some papers, let me phrase it this way.

Given a family of groups $\{G_{k}\}$, say $G_{k} \subset G_{k+1}$, (what I have in mind are symmetric groups, braid groups, mapping class groups, general linear group, etc) let $M := \coprod_{k \geq 0} BG_k$, and $G_{\infty} = $ direct limit of $G_k$.

Is it true that the group completion of $M$, is $\mathbb{Z} \times BG_{\infty}^{+}$? Maybe we need some additional structure for $M$ and the groups $G_k$ - like, that they form a monoidal category or something....?

Basically, what are the conditions so that the group completion of $M$ is, $\mathbb{Z} \times BG_{\infty}^{+}$?