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 <i> space of path lifting problems </i>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 <i> parallel transport </i> 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 of fibers.)


Evaluating the section when $x=y$ is the basepoint gives the holonomy operation 
$\Omega B \times F \to F$, or adjointly as $\Omega B \to G(F)$, where $G(F)$ is
the topological monoid of self homotopy equivalences of $F$. If $p$ is a fiber bundle with structure group $G$, then
the transport operation described above can be factored as 
$$
\Omega B \to G\to G(F) .
$$
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.