  We derive the secular evolution of the orbital elements of a stellar-mass
object orbiting a spinning massive black hole. We use the post-Newtonian
approximation in harmonic coordinates, with test-body equations of motion for
the conservative dynamics that are valid through 3PN order, including
spin-orbit, quadrupole and (spin)$^2$ effects, and with radiation-reaction
contributions linear in the mass of the body that are valid through 4.5PN
order, including the 4PN damping effects of spin-orbit coupling. The evolution
equations for the osculating orbit elements are iterated to high PN orders
using a two-timescale approach and averaging over orbital timescales. We derive
a criterion for terminating the orbit when its Carter constant drops below a
critical value, whereupon the body plunges across the event horizon at the next
closest approach. The results are valid for arbitrary eccentricities and
arbitrary inclinations. We then analyze numerically the orbits of objects
injected into high-eccentricity orbits via interactions within a surrounding
star cluster, obtaining the number of orbits and the elapsed time between
injection and plunge, and the residual orbital eccentricity at plunge as a
function of inclination. We derive an analytic approximation for the time to
plunge in terms of initial orbital variables. We show that, if the black hole
is spinning rapidly, the flux of gravitational radiation during the final orbit
before plunge may be suppressed by as much as three orders of magnitude if the
orbit is retrograde on the equatorial plane compared to its prograde
counterpart.
