  We compute the gravitational radiation-reaction force on a compact binary
source at the fourth-and-a-half post-Newtonian (4.5PN) order of general
relativity, i.e., 2PN order beyond the leading 2.5PN radiation reaction. The
calculation is valid for general orbits in a general frame, but in a particular
coordinate system which is an extension of the Burke-Thorne coordinate system
at the lowest order. With the radiation-reaction acceleration, we derive (from
first principles) the flux-balance laws associated with the energy, the angular
and linear momenta, and the center-of-mass position, in a general frame and up
to 4.5PN order. Restricting our attention to the frame of the center of mass,
we point out that the equations of motion acquire a non-local-in-time
contribution at the 4.5PN order, made of the integrated flux of linear momentum
(responsible for the recoil of the source) together with the instantaneous flux
of center-of-mass position. The non-local contribution was overlooked in the
past literature, which assumed locality of the radiation-reaction force in the
center of mass frame at 4.5PN order. We discuss the consequences of this
non-local effect and obtain consistent non-local equations of motion and flux
balance laws at 4.5PN order in the center-of-mass frame.
