  The "flux-balance formulae" that determine the averaged evolution of energy,
azimuthal angular momentum, and Carter constant in terms of the averaged
asymptotic gravitational-wave fluxes for inspirals of small bodies into Kerr
black holes were first derived about 15 years ago. However, this derivation is
restricted to the case that the background Kerr geodesics are non-resonant
(i.e., the radial and angular motions are always incommensurate), and excludes
the resonant case that can be important for the radiative dynamics of extreme
mass-ratio inspirals. We give here a new derivation of the flux formulae based
on Hamiltonian dynamics of a self-forced particle motion, which is a valuable
tool for analyzing self-force effects on generic (eccentric, inclined) bound
orbits in the Kerr spacetime. This Hamiltonian derivation using action-angle
variables is much simpler than the previous one, applies to resonant inspirals
without any complication, and can be straightforwardly implemented by using
analytical/numerical Teukolsky-based flux codes.
