For a Serre fibration of pointed topological spaces $f:X \to B$, there is an action of $\pi_1\left(B,b_0\right)$ on the fiber $F$. The construction of this action I'm familiar with uses a lift $F\times I \to X$ of the map $F \times I \xrightarrow{\pi} I \xrightarrow{\gamma} B$ for any $\left[\gamma \right] \in \pi_1 \left(B,b_0 \right)$.

Now for a general map between two $\infty$-groupoids $f:X \to B$, we can use some version of the Grothendieck construction to construct a map $\phi_f : B \to \operatorname{Grp}_\infty$, and then for an element $\left[\gamma \right] \in \pi_1 \left(B,b_0 \right)$, $\phi_f \left(\gamma \right)$ is an automorphism of $\phi_f \left(b_0 \right)$, which, I guess, generalizes the previous definition (please correct me if that is already false). Is there an explicit description of this automorphism for a specific $\gamma$? (i.e. in terms of pullbacks / pushouts / sections of the maps $f,\gamma$ etc.) I'm particularly interested in writing down obstructions for triviality of this action.