This question ought to have a straightforward (perhaps even glaringly obvious) answer, but so far I've already wasted a few hours trying to untangle this web of inconsistent identifications. I'm sure someone more experienced in this area will be able to quickly point out exactly where things go wrong.

Let $G$ be a Lie group and let $\mathbf{B} G$ denote the moduli stack obtained by fixing a model for the homotopy orbits of $G$ acting on the point (For example, one can take the nerve of the resulting action groupoid).

This smooth stack classifies smooth $G$-bundles. More precisely, if we are given smooth manifold $M$, the mapping space ${\rm Map}(M,\mathbf{B}G)$ is equivalent to the infinity groupoid ${\rm Bun}^{\rm sm}_G(M)$ of smooth principal $G$-bundles on $M$, with smooth isomorphisms and higher isomorphisms between them (To prove this one can, for example, resolve $M$ by its Cech nerve and identify the resulting mapping space directly).

I was under the impression that the map ${\rm Bun}_G^{\rm sm}(M)\to {\rm Bun}_G(M)$, which forgets the smooth structure and returns the underlying (topological) principal $G$-bundle was a natural equivalence. If this is the case, then since the latter is equivalent to ${\rm Map}(M,BG)$ and the projection $M\times \mathbb{R}\to M$ induces an equivalence $${\rm Map}(M,BG)\to {\rm Map}(M\times \mathbb{R},BG)\;,$$ we should conclude that $\mathbf{B}G$ is homotopy invariant.

On the other hand, if $\mathbf{B}G$ were homotopy invariant, then it should be equivalent to the locally constant stack of its global sections. The global sections functor defines a right infinity adjoint and should therefore commute with delooping. But the global sections of $G$ just return the points of $G$, viewed as a discrete simplicial set. We would therefore be forced identify $\mathbf{B}G$ with the locally constant stack of $BG^{\delta}$, where $\delta$ indicates that we have taken $G$ to have the discrete topology. This stack clearly classifies something very different. In particular, if $M$ is a smooth manifold, then we can identify the space of maps to this stack with the space ${\rm Map}( M ,BG^{\delta})$, where we have forgetten the smooth structure on $M$.

**Question**

Clearly, something is wrong here. Can someone help point me in the right direction?