I know that when $f_0: X \to Y$ and $f_1: X \to Y$ are homotopic maps between paracompact manifolds, and $E \to Y$ is a vector bundle or principal bundle with finite-dimensional fibres over $Y$, that $f_0^* E \cong f_1^* E$ as bundles over $X$.
I am curious what are the most general circumstances under which this holds? For instance, how general can we allow the spaces $X$ and $Y$ to be? Does this theorem still hold for any fibre bundle $E \to Y$ with say infinite-dimensional fibres or infinite-dimensional structure group?