In Haynes Miller's "Notes on Cobordism" (pgs 39-40), he says the following:
Suppose we have a directed system of topological groups $G_0 \rightarrow G_1 \rightarrow \dots$ along with a compatible system of orthonormal representations $G_i \rightarrow O(i)$. Then if $G$ denotes the direct limit, applying the classifying space functor to the induced map $G \rightarrow O$ results in a fibration $BG \rightarrow BO$.
It seems to me like there is no reason this should be a fibration. For example, we could let all of our representations be trivial. Is there some condition on the representations that should be added?