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For questions about or involving fibrations which are maps which satisfy the homotopy lifting property for all spaces.
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A conceptual proof that local fibrations over paracompact spaces are global fibrations?
locally-finite cover $V\to B$ such that the pullback
$$\begin{array}{ccc} E' & \rightarrow & E \\\\ \downarrow & & \downarrow \\\\ V & \rightarrow & B\end{array} $$
has $E'\rightarrow V$ a Hurrewicz fibration … to V$, hoping they glue together properly; which, admitedly, was the problem to start with; only now it looks more hopeful: in particular, I expect we can claim a lemma that for $E'\to V$ a Hurrewicz fibration …