For which G is BLG weak homotopy equivalent to LBG? Let $G$ be a (Edit: path-)connected topological group. Under what additional hypotheses on $G$ is it true that $LBG$ is a classifying space for $LG$? (or, I guess equivalently, when is $LBG \sim BLG$?) Here I'm taking the free loop space, and the compact-open topology on it. I know it is true for strong hypotheses, such as $G$ a compact Lie group. If one can find a model of $BG$ that is locally contractible and paracompact, then by Atiyah and Bott's The Yang-Mills Equations over Riemann Surfaces (doi:10.1098/rsta.1983.0017), Proposition 2.4, I believe it is possible. So, alternatively, for what $G$ is it true that $BG$ can be thus chosen?
 A: [UPDATE: There were some mistakes in the first version.  Here is a more careful account.]
I'll work everywhere with CGWH spaces, so I have a Cartesian closed category.
Note that $BLG$ is always path-connected, but $\pi_0(LBG)=\pi_0(G)/\text{conjugacy}$, so we need to assume that $G$ is path-connected.  (The question says connected, but this may be a bit stronger; I do not know whether there are connected topological groups that are not path-connected.)  
For any space $X$ and any $t\in S^1$ we have an evaluation map $\epsilon_{X,t}\colon LX\to X$.  For the special case of the basepoint $1\in S^1$ we write $p_X=\epsilon_{X,1}\colon LX\to X$.  This is always a Hurewicz fibration.  If $X$ is based we have a fibre sequence $\Omega X\xrightarrow{i_X}LX\xrightarrow{p_X}X$.
Now let $G$ be a topological group.  We write $EG$ and $BG$ for the usual simplicial constructions, so $EG$ is contractible and has a free $G$-action with orbit space $BG$.  Simplicial methods also give a natural commutative diagram 
$\require{AMScd}$
\begin{CD}
G @>k_{G}>> EG @>r_{G}>> BG\\
@Vj_GVV @VV l_G V @VV 1 V\\
\Omega BG @>>> PBG @>>> BG
\end{CD} 
Both $EG$ and $PBG$ are contractible, and the bottom row is a Hurewicz fibration.  If the top row is also a Hurewicz fibration, we can conclude that $j_G\colon G\to\Omega BG$ is a homotopy equivalence.  If the top row is merely a Serre fibration or quasifibration, we can still conclude that $j_G$ is a weak equivalence.  I do not know what are the minimal conditions for the top row to be a quasifibration.  
Next, given $u\in BLG$ and $t\in S^1$ we have a homomorphism $\epsilon_{G,t}\colon LG\to G$ and thus a map $B\epsilon_{G,t}\colon BLG\to BG$ and thus an element $(B\epsilon_{G,t})(u)\in BG$.  We would like to define $f_G\colon BLG\to LBG$ by $(f_G(u))(t)=(B\epsilon_{G,t})(u)$.  To justify this we need to check continuity in $t$ and then in $u$.  This in turn needs some continuity properties of the functor $B$, which can be proved using some abstract nonsense with CGWH spaces and Cartesian closure.  (The initial version of this answer referred to a natural map in the opposite direction, but I think that does not actually exist.)  We now want to construct a diagram as follows:
\begin{CD}
 B\Omega G @>Bi_G>> BLG @>Bp_G>> BG \\
 @V f'_G VV @V f_G VV @VV 1 V \\
 \Omega BG @>>i_{BG}> LBG @>>p_{BG}> BG 
\end{CD}
Constructions that we have already discussed provide all spaces and maps except for $f'_G$.  On the top row we note that $(Bp_G)\circ (Bi_G)$ is trivial, and on the bottom row we know that $i_{BG}$ is the fibre of $p_{BG}$, so there is a unique way to fill in $f'_G$.  The bottom row is always a Hurewicz fibration.  The top row is obtained by applying $B$ to a Hurewicz fibration of topological groups, but it is not clear exactly what we get from that.  If the top row is at least a Serre fibration, we see that $f_G$ is a weak equivalence iff $f'_G$ is a weak equivalence.
Finally, define $\tau\colon S^1\wedge S^1\to S^1\wedge S^1$ by $\tau(s\wedge t)=t\wedge s$.  From the definitions one can check that the following diagram commutes:
\begin{CD}
 \Omega G @>j_{\Omega G}>> \Omega B\Omega G \\
 @V \Omega j_G VV @VV \Omega f'_G V \\
 \Omega^2 BG @>>\tau^*> \Omega^2 BG 
\end{CD}
If $j_G$ and $j_{\Omega G}$ are weak equivalences, we conclude that $f'_G$ is also a weak equivalence.
All this assumes that we start with the simplicial definition of $BG$.  One could instead consider an axiomatic characterisation of $BG$, which might include the condition that the map $EG\to BG$ is a Serre fibration.  The idea should be that $[X,BG]$ should biject with the set of isomorphism classes of principal $G$-bundles over $X$, but one would need to restrict attention to paracompact $X$, or to principal bundles over arbitrary $X$ that admit a numerable trivialising cover.  I do not know how the technicalities would work out.
