For a Lie group $G$ let $EG \to BG$ denote the universal bundle. A Lie group homomorphism $\rho: G \to H$ determines a map $B \rho: BG \to BH$ as the classifying map for the principal $H$-bundle $EG \times_\rho H \to BG$. While this argument yields existence of $B \rho$ (and uniqueness up to homotopy) it is far from being explicit. Does there exist a true construction of $B \rho$? If $\rho$ is a embedding of a subgroup, then on gets the classifying map by a quotient procedure. I was hoping for a similar result for arbitrary group homomorphism. Assume anything you want on $H$ (compactness ect).

Some background motivation: It is well-known that universal bundles characterize equivalence classes of principal bundles. A further, not so common characterization is given by 'homomorphisms $\rho$' between the loop space of $M$ and $G$. The principal bundle $P$ is then the to the path bundle associated bundle $PM \times_\rho G$. Hence there are two classification of principal bundles with connections and I want to better understand the interplay between them.