When does P-localization preserve homotopy pushouts or pullbacks?

We give conditions under which localization at a set of primes P in the sense of Casacuberta and Peschke [Trans. Amer. Math. Soc. 339 (1993) 117–140] preserves homotopy pushouts and homotopy pullbacks. We then apply these results to infer conditions under which P-localization preserves homotopy epimorphisms and homotopy monomorphisms. We also obtain conditions under which P-localization of non-nilpotent spaces induces P-localization of its homotopy groups.