Let's consider all flat fiber bundles with base space $B$ and fiber $F$, where $B$ and $F$ are compact and at least CW-complexes. (perhaps even topological/smooth manifolds if that helps)
All those bundles are of the form $$E \cong \widetilde{B} \times_{\pi_{1}} F,$$ where $\widetilde{B}$ is the universal cover of $B$, $\pi_{1}$ is the fundamental group of $B$ and $F$ carries a $\pi_{1}$-action.
Now the question is: How can I see, given two $\pi_{1}$-actions on $F$, whether the corresponding bundles are fiber-wise homotopy equivalent/homeomorphic. Are there corresponding equivalence relations on the set (space?) of $\pi_{1}$-actions on $F$?
Example
Let $B=F=S^1$. Then there are exactly two corresponding flat bundles, namely the trivial one and the klein bottle.
Since $\pi_{1}=\mathbb{Z}$, we can identify $\pi_{1}$-actions on $F$ with automorphisms of $F$ (any $\mathbb{Z}$-action is determined by the action of $1$). The latter consists of exactly two connected components (at least this is true for self-diffeomorphisms), where one reverses orientation and the other doesn't.
Now, the orientation-reversing component corresponds to the klein-bottle while the the other corresponds to the trivial bundle.
Can this be generalized?
