For a topological group $G$, an older term for a map $BG \to BG$ is a "homotopy homomorphism". If $G$ is connected, taking based loops shows that a homotopy class of such a map is the same as a homotopy class of $E_1$-maps $G \to G$.
For a compact connected simple Lie group $G$, the homotopy classes of maps $BG \to BG$ were classified by Stefan Jackowski, James McClure and Bob Oliver: by Theorem 2 of Homotopy Classification of Self-Maps of BG via G-actions all are given by composing a generalised Adams operation with a map induced by a homomorphism $G \to G$.
I am interested in analogous results for the group $G = \mathrm{Top}(d)$ of homeomorphisms $\mathbb{R}^d \to \mathbb{R}^d$ in the compact-open topology:
What is known about the homotopy classes of maps $B\mathrm{Top}(d) \to B\mathrm{Top}(d)$?
