Let $X$ be a paracompact space and consider two homotopic maps $f,g: X\to B$, where $E\to B$ is a real vector bundle over a paracompact space $B$. We know that these maps have isomorphic pullback bundles $E'\to X$, where we can assume that the corresponding maps $f',g': E'\to E$ are metric-preserving.
Are the induced maps of Thom spaces $Th(f),Th(g): Th(E')\to Th(E)$ also homotopic?