This is a crosspost of this MSE question.

Let $A\overset{\alpha}{\rightarrow}B$ be a (Hurewicz) fibration.

- The homotopy lifting property w.r.t a fiber $\alpha ^{-1}(b)$ furnishes for each path $b\to b^\prime$ in the base a continuous map $\alpha ^{-1}(b)\to \alpha ^{-1}(b^\prime)$. Moreover, this assignment extends to a functor $\pi_1B\longrightarrow \mathsf{hTop}$.
- On the other hand, as a fibration $\begin{smallmatrix}A\\\downarrow\\B\end{smallmatrix}$ admits a Hurewicz connection $s$. Given such a connection it is tempting to send a path $b\overset{\gamma}{\to} b^\prime $ in the base to the following set function (analogously to covering space theory) $$\alpha^{-1}(b)\longrightarrow \alpha ^{-1}(b^\prime),\quad a\mapsto \operatorname{eval}_1s(a,\gamma).$$ I suspect this set function might be continuous, but I see no reason for it to be a homotopy equivalence, since $s(a,\gamma)$ need not be related in a nice way to lifts of opposite path $b\overset{\bar\gamma}{\leftarrow} b^\prime $.

**Questions.**

- Is fiber transport along a Hurewicz connection continuous?
- (Assuming continuity) Is fiber transport along a Hurewicz connection functorial?
- (Assuming continuity) Does it coincide with the first transfer functor?
- Suppose the fibers are all homeomorphic. Are there any interesting conditions that make the transport functor $\pi_1B\longrightarrow \mathsf{hTop}$ lift to $\mathsf{Top}$? That is, can we obtain such homeomorphisms via transport?

I thought about possible functoriality of the transport along the Hurewicz connection. We want $\operatorname{eval}_1(a,\delta \ast \gamma)=\operatorname{eval}_1s(\operatorname{eval}_1s(a,\gamma),\delta,)$. We may consider the concatenation $s(\operatorname{eval}_1s(a,\gamma),\delta)\ast s(a,\gamma)$ which seems to lift $\delta \ast \gamma$, but I'm not quite sure where to go from here.