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).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 $\Omega B \to \text{homeo}(F)$. 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 $$\Omega B \to G\to \text{homeo}(F) G(F) .$$ 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$p\: p: EG \to BG$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. 3 added 153 characters in body 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$E$is contractiblep\: EG \to BG$ and $F = G$, the then map $\Omega B BG \to F$ G$will be a homotopy equivalence, and using the above homotopy fiber sequence, since$E = EG$is contractible. We have also seen this map is as decribed by the orbit of a point in$G$under the holonomy operation$\Omega BG \times G \to G$as given above. 2 added 1555 characters in body Something like this I'm going to change my original answer, since I interpreted the question wrongly. (I hope that's alright.) Suppose$p: E\to B$is worked out a Hurewicz fibration, where$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$which might be called a "holonomy" in papers of Pawel Gajerthe algbraic topology sense. The idea is this: Gajer, Paweł Higher holonomies, geometric loop groups Let \Lambda_p = E\times_B B^Ibe the space of path lifting problems for$p$(this is the space of pairs$(e,\lambda)$where$e\in E$and smooth Deligne cohomology$\lambda$is a path starting at$p(e)$. There is a mapq: E^I \to \Lambda_pby sending path$\lambda$in$E$to$(\lambda(0), p\circ \lambda)$. Then the condition that$p$be a Hurewicz fibration is tantamount to saying that$q$has a section. Advances A choice of section might be regarded as {\it parallel transport} along a path in geometry, 195–235, Progrthe algebraic topological sense. Math., 172, Birkhäuser Boston, Boston, MAChoose such a section. This gives a way of associating to eachpath in$B$, starting at$x$and ending at$y$, a map$E_x \to E_y$, 1999where$E_x$isthe fiber at$x$. This map is a homotopy equivalence. (I suggest you also see its Math Review) The rough idea When$p$is a fiber bundle,one can choose the section in such a way that there's each parallel transport is a bona-fide topological group model for homeomorphism). Evaluating the loop space section when$x=y$is the basepoint gives the holonomy operation$\Omega M$when B \times F \to F$, or adjointly as $M$ \Omega B \to \text{homeo}(F)$. If$p$is smooth and compacta fiber bundle with structure group$G$, thenthe transport operation described above can be factored as \Omega B \to G\to \text{homeo}(F) .Holonomy is $$If we choose a basepoint in F, then the value of the operation on the basepoint gives a bona-fide topological group homomorphism map\Omega B \to F \, .This map is well-known: it's the map sitting in the homotopy fiber sequence\Omega B \to F \to E .$$ (this should be in any reasonable text on the subject). So, in the particular case when$E$is contractible, the map$\Omega M B \to G$F$ will bea homotopy equivalence, and this map is decribed by orbit the holonomy operation as given above.