I am reading about the generalization of the monodromy action to fibrations.
In this settings, the lift of a path is not unique, but it is unique up to free homotopy. In particular there is a well defined right action of $\pi_1(X,x)$ on $\pi_0(F_x)$ ($F_x$ is the fiber over $x$) which sends a path component of the fiber and a homotopy class of loops, to the path component of the ending point of a lift of a path representing the class in $\pi_1(X,x)$.
Of course if $F_x$ is discrete, then $\pi_0(F_x) \cong F_x$ and we recover the usual monodromy for coverings.
This got me thinking about the following construction.
Let $p: E \to X$ be a fibration (not sure which hypothesis you need). Consider the set: $$ \pi_0^v(E) := \bigsqcup_{x \in X} \pi_0(p^{-1}(x))$$ with the function $\pi_0^v(p) : \pi_0^v(E) \to X$ defined by $$\pi_0^v(p)|_{\pi_0(p^{-1}(x))} : [e] \mapsto x$$ I used a superscript $v$ for "vertical".
By putting some topology on $\pi_0^v(E)$ (quotient topology?) we could end up with a covering map. At this point, for example, the two notion of monodromy action will coincide.
This construction might be a functor from some category to the category of coverings.
Questions: Is this construction known/used? (First: Does it actually work?)
I am not asking to provide full details (but of course they are gladly accepted). I am mainly interested in knowing if this is known to work and used somewhere in mathematics.
In general your construction does not work.
Consider fibrations with unique path lifting. These are more general than covering maps. A nice treatment an be found in
It is easy to see that a fibration has unique path lifting if and only each fiber has only constant paths. In that case $\pi_0(p^{-1}(x))$ can be naturally identified with $p^{-1}(x)$ so that $\pi_0^v(E)$ is the same set as $E$. Thus the quotient map $E \to \pi_0^v(E)$ is a bijecton and (if you are not willing to give $\pi_0^v(E)$ an arbitrary topology) you will agree that the only reasonable topology on $\pi_0^v(E)$ is that making $E \to \pi_0^v(E)$ a homeomorphism.
Now there are fibrations with unique path lifting which are no covering maps, for example the infinite product $\Pi_{n=1}^\infty p_n : \Pi_{n=1}^\infty\mathbb R \to \Pi_{n=1}^\infty S^1$, where each $p_n(t) = e^{2\pi it}$ (see Example 9 in Spanier, Ch. 2 Sec. 2). For such maps your construction does not produce a covering map. Note that both $\Pi_{n=1}^\infty\mathbb R$ and $\Pi_{n=1}^\infty S^1$ are nice in the sense that they are path connected and locally path connected.
In this context also have a look at Theorem 10 in in Spanier, Ch. 2 Sec. 4 which says that if $p : E \to B$ is a fibration with unique path lifting such that $B$ is locally path connected and semilocally $1$-connected and $E$ is locally path connected, then $p$ is a covering map.