Air mixing systems for gas turbine engines include a heat exchanger, a first extraction conduit fluidly coupled to an inlet of the heat exchanger, a second extraction conduit fluidly coupled to an outlet of the heat exchanger, an injection conduit fluidly coupled to the second extraction conduit, an onboard injector supply chamber fluidly coupled to the injection conduit, and an onboard injector fluidly coupled to the onboard injector supply chamber.