I'm going to change my original answer, since I interpreted the question wrongly. (I hope that's alright.)
Suppose $p: E\to B$ is a Hurewicz fibration, where $F = p^{-1}(\ast)$ is the fiber over the basepoint and $B$ is connected. Then one can cook up a map $\Omega B \times F \to F$ which might be called a "holonomy" in the algbraic topology sense.
The idea is this: Let $$ \Lambda_p = E\times_B B^I $$ be the space of path lifting problems for $p$ (this is the space of pairs $(e,\lambda)$ where $e\in E$ and $\lambda$ is a path starting at $p(e)$. There is a map $$ q: E^I \to \Lambda_p $$ by sending path $\lambda$ in $E$ to $(\lambda(0), p\circ \lambda)$. Then the condition that $p$ be a Hurewicz fibration is tantamount to saying that $q$ has a section. A choice of section might be regarded as {\it parallel transport} along a path in the algebraic topological sense. Choose such a section. This gives a way of associating to each path in $B$, starting at $x$ and ending at $y$, a map $E_x \to E_y$, where $E_x$ is the fiber at $x$. This map is a homotopy equivalence. (When $p$ is a fiber bundle, one can choose the section in such a way that each parallel transport is a homeomorphism).
Evaluating the section when $x=y$ is the basepoint gives the holonomy operation $\Omega B \times F \to F$, or adjointly as $\Omega B \to \text{homeo}(F)$. If $p$ is a fiber bundle with structure group $G$, then the transport operation described above can be factored as $$ \Omega B \to G\to \text{homeo}(F) . $$ If we choose a basepoint in $F$, then the value of the operation on the basepoint gives a map $$ \Omega B \to F \, . $$ This map is well-known: it's the map sitting in the homotopy fiber sequence $$ \Omega B \to F \to E . $$ (this should be in any reasonable text on the subject).
So, in the particular case when $p\: EG \to BG$ and $F = G$, then map $\Omega BG \to G$ will be a homotopy equivalence, using the above homotopy fiber sequence, since $E = EG$ is contractible. We have also seen this map as decribed by the orbit of a point in $G$ under the holonomy operation $\Omega BG \times G \to G$ as given above.