Suppose we are given a diagram $X \to Z \gets Y$ of $G$spaces ($G$ a discrete group). Let $( \times^h )$ denote homotopy pullback. Is $(X \times^h_Z Y)_{hG}$ weakly equivalent to $X_{hG} \times^h_{Z_{hG}} Y_{hG}$?
Yes. A sketch: Taking products with the free $G$space $EG$ commutes with the pullback diagram (because product is also a limit) and so you can assume they're free, and one of the maps is a fibration. Having done this, there is a natural long exact sequence of homotopy groups $\to \pi_* (U) \to \pi_*(U_{hG}) \to \pi_*(BG) \to \dots$ and applying this to the pullback diagram you can deduce (from the 5lemma) that the natural map from the orbit of pullbacks to the pullback of the orbits is a weak equivalence. 

