There's a construction of $BG$ used quite often by geometric-oriented homotopy theorists that makes the homotopies quite explicit. In this model for $BG$ you start by "resolving" $G$ to the homotopy-equivalent space of finite subsets of $[0,1]$ labelled by elements of $G$, with the rules that if two labelled points collide then their labels multiply. To make this definition precise you have to take the disjoint union $\sqcup_n \Delta_n \times G^n$ and mod out by the collision relation, given by maps of the boundary facets to $\Delta_{n-1} \times G^{n-1}$. Here $\Delta_n = \{ (t_1,\cdots, t_n) : 0 \leq t_1 \leq \cdots \leq t_n \leq 1 \}$ is the $n$-simplex. You similarly allow for the deletion of points labelled by the neutral element.
In this model there are two actions of $G$ on this "resolved" $G$, one by inserting a point on the left, and another by inserting a point on the right. You mod out by one of these actions and you have a model for $EG$, and mod out by both to get $BG$.
Anyhow, so say you have a homomorphism $f : G \to H$, then the induced map of classifying spaces $Bf$ simply takes a collection of $G$-labelled points in $[0,1]$ and replaced them by the $H$-labelled points by applying $f$ to the labels. So if all the labels are conjugated $x^{-1} f(g) x$ by a common element you can slide the leftmost $x^{-1}$ off the $0$ end of the interval, the rightmost $x$ off the $1$ end of the interval, and all the other $x^{-1}$ and $x$ pairs can slide together and cancel. That would be the homotopy.
Dev Sinha gives the appropriate images to model this in his presentation: https://www.youtube.com/watch?v=epxkF_LDjrQ
it appears around the 18-minute mark. But he goes into detail in the case of $\mathbb RP^\infty$. Which might be what you are looking for.