I wanted to ask a summary of known results and references about the homotopy type of the mapping space $\mathrm{Map}(BG,BK)$ (and specially the connected components) between the classifying spaces when G and K are general topological groups. Thank you.

As Neil Strickland points our, this is too difficult to have a good answer in general. In this post, I am only considering discrete groups. A very special case is the following
Examples of superpositions include retractions, maps between oriented manifolds of nonzero degree, maps whose homotopy fibre is a finite complex with nonzero Euler characteristic. In the application below, one needs a much weaker condition which might be checked by hand for particular maps. The following result can be proved:
This is (a special case of) a consequence of results in [Daniel H. Gottlieb. Covering transformations and universal fibrations. Illinois J. Math., 13:432–437, 1969., Daniel H. Gottlieb. Self coincidence numbers and the fundamental group. J. Fixed Point Theory Appl. 2 (2007), no. 1, 73–83.] and was proved in [Thomas Schick and Andreas Thom. On a conjecture of Gottlieb. Algebr. Geom. Topol. 7 (2007), 779–784.] 


The Sullivan conjecture, a beautiful theorem proved by Haynes Miller (The Sullivan conjecture on maps from classifying spaces) has striking consequences about maps between classifying spaces when the target is completed at a prime $p$. Some references are Dwyer and Zabrodsky (Maps between classifying spaces), Jackowski, McClure, and Oliver Homotopy classification of selfmaps of $BG$ via $G$actions. I, II; Selfhomotopy equivalences of classifying spaces of compact connected Lie groups); Notbohm (Maps between classifying spaces). The classical work of J.F. Adams on this topic is also well worth remembering: Maps between classifying spaces (I), II, III and Maps between $p$completed classifying spaces. 

