For any group $G$, the universal example for proper $G$-actions, $\underline{E}G$, is a proper $G$-space such that for any other proper $G$-space $X$, there exists a map (unique up to $G$-equivariant homotopy)
$$X\to\underline{E}G.$$
The quotient $\underline{B}G$ may be called the classifying space for proper $G$-actions.
Question 1: Does a homomorphism of groups $f\colon G\to H$ induce a $G$-equivariant continuous map $f_*\colon\underline{E}G\to\underline{E}H$ and hence also a continuous map $\underline{B}G\to\underline{B}H$?
Question 2: Under what conditions do we have that $\underline{B}(G_1\times G_2)$ is homotopy equivalent/homeomorphic to $\underline{B}G_1\times\underline{B}G_2$?
Best Answer
Here is model that is obviously functorial: take for $\underline{E}G$ the simplicial complex with vertex set the finite subsets of $G$ and simplices the finite chains of sets ordered by inclusion. A homomorphism $f:G\rightarrow H$ induces a function from the finite subsets of $G$ to the finite subsets of $H$. You can also think of this model as the barycentric subdivision of the `simplex' with vertex set $G$.
I can't think of anything to add to Fernando Muro's answer to question 2 in the comments.