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 $X\_{hG} \times^h\_{Z\_{hG}} Y\_{hG}$?



Do homotopy pullbacks commute with homotopy orbits (in spaces)?
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 5-lemma) that the natural map from the orbit of pullbacks to the pullback of the orbits is a weak equivalence.


