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.



Action of fundamental group on homotopy fiber
(This answer is written in a model-independent fashion -- translate to your favourite formalism).


For every path $\gamma:[0,1]\to B$ you get an isomorphism in the homotopy category $X\_{\gamma0}\xrightarrow{\sim} X\_{\gamma1}$ (where with $X\_b$ I denote the homotopy fiber over $b\in B$). Probably the easiest and most geometric way of constructing it is to consider the space of lifts.


Let $\operatorname{Sec}\_\gamma(f)$ be the space whose objects are sections up to homotopy over $\gamma$
$$\operatorname{Sec}\_\gamma(f)=\{(\tilde\gamma,H)\mid \tilde\gamma:[0,1]\to X,\ H:f\tilde\gamma\sim \gamma\}$$
(this is nothing else but the homotopy fiber over $\gamma$ of the map $X^{[0,1]}\to B^{[0,1]}$). Then you have a zig zag of maps
$$ X\_{\gamma0}\xleftarrow{ev\_0} \operatorname{Sec}\_\gamma(f)\xrightarrow{ev\_1} X\_{\gamma1}$$
where the two maps are evaluation at 0 and 1 respectively. Both maps are homotopy equivalences (this requires some proof, but it's not terribily hard: they're both trivial fibrations when $f$ is a fibration), and so you can define the map in the homotopy category as $ev\_1 \circ ev\_0^{-1}$.


In order to prove that the action of a loop is trivial, you'll have to prove that $ev\_0$ and $ev\_1$ are homotopic. I'm not aware of a general way of attacking this problem, but of course studying the behaviour of the two maps on various algebraic invariants can often provide obstructions.


