  The motion of an extended, but still weakly gravitating body in general
relativity can often be determined by a set of conserved quantities. If they
are sufficient in number, this allows for a complete solution to the equations
of motion, much like for geodesic motion. Under the gravitational self-force
(relaxing the "weakly gravitating" assumption), these "conserved quantities"
evolve with time. This evolution can be calculated using the (local) self-force
on the body, but such an approach is computationally intensive. To avoid this,
one often instead uses flux-balance laws: relationships between the average
evolution (capturing the dissipative dynamics) and the values of the field far
away from the body, which are far easier to compute. In the absence of spin,
such a flux-balance law has been proven in [Isoyama et al., 2019] for any of
the conserved action variables appearing in a Hamiltonian formulation of
geodesic motion in the Kerr spacetime. In this paper, we attempt to extend this
flux-balance law to linear order in spin, in the context of an arbitrary black
hole spacetime, and in a way that directly relates the average rates of change
to the flux of a conserved current through the horizon and to infinity. In the
absence of spin, we reproduce results consistent with those in [Isoyama et al.,
2019]. To linear order in spin we show that most, but not all, constants of
motion at linear order in spin appear to possess flux-balance laws.
