Choose a notion of an "ambient homotopy" between maps of topological spaces. For example, say that two embeddings $Y \rightarrow X$ are ambiently homotopic if there is a path between them in the space of embeddings from $Y$ to $X$.

Q. Given a fiber bundle $p: E \rightarrow B$, and two ambiently homotopic maps $f, g : X \rightarrow B$, are pullbacks $X_f \rightarrow E$ and $X_g \rightarrow E$ of $f$ and $g$ along $p$ ambiently homotopic (modulo somehow identifying $X_f$ with $X_g$)?

If one takes $X$ to be the space with two disjoint points, then $X_f$ and $X_g$ both are two disjoint copies of the fiber of $p$, so they are homeomorphic. Using some homeomorphism to identify $X_f$ and $X_g$ and using the above definition of the ambient homotopy, what is the answer to the question?

For example, consider the Hopf bundle $S^3 \rightarrow S^2$. Take $X$ to be the two point set. Then the two pullpacks $X_f \rightarrow S^3$ and $X_g \rightarrow S^3$ will both be two linked circles in $S^3$. These are "ambiently homotopic".

More generally, how to define the appropriate notion of an "ambient homotopy" so that the question makes sense?

fandgalongphave different codomains $f^*E$, $g^*E$. To make sense of the question, one has to identify these somehow; one certainly can do so, since they’re pullbacks of homotopic maps into a bundle, but the answer may be sensitive to how one does so, and how nice your notion of ambient homotopy is. $\endgroup$ – Peter LeFanu Lumsdaine Nov 17 '15 at 0:26homotopically equivalent. This means there is a homotopy equivalence $h: Y\to Y'$ making the evident triangle commute. I think that is the best you can hope for here. $\endgroup$ – Mark Grant Nov 17 '15 at 10:06