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 homeomorphic to $\underline{B}G_1\times\underline{B}G_2$?