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Let $G$ be a discrete group. Let $P$ denote the groupoid whose objects are the elements of $G$, and whose morphisms from $a$ to $b$ are the elements $g\in G$ such that $gag^{-1}=b$. It is then a standard fact that $\Omega'BG=BP$. Less naturally but more concretely, if we choose a set $C$ of representatives for the conjugacy classes in $G$, we get an equivalence $\Omega'BG=\coprod_{c\in C}BZ_G(c)$. If $G$ is nonabelian then this will be different from $BG\times \Omega BG=BG\times G$. For example, you could take $G$ to be the free group on two generators, so $BG$ is a figure eight.