  We compute the leading Post-Newtonian (PN) contributions at linear order in
the spin to the radiation-reaction acceleration and spin evolution for binary
systems, which enter at fourth PN order. The calculation is carried out, from
first principles, using the effective field theory framework for spinning
compact objects, in both the Newton-Wigner and covariant spin supplementary
conditions. A non-trivial consistency check is performed on our results by
showing that the energy loss induced by the resulting radiation-reaction force
is equivalent to the total emitted power in the far zone, up to so-called
"Schott terms." We also find that, at this order, the radiation reaction has no
net effect on the evolution of the spins. The spin-spin contributions to
radiation reaction are reported in a companion paper.
