  We extend the Ori and Thorne (OT) procedure to compute the transition from an
adiabatic inspiral into a geodesic plunge for any spin, with emphasis on
near-extremal ones. Our analysis revisits the validity of the approximations
made in OT. In particular, we discuss possible effects coming from eccentricity
and non-geodesic past-history of the orbital evolution. We find three different
scaling regimes according to whether the mass ratio is much smaller, of the
same order or much larger than the near extremal parameter describing how fast
the primary black hole rotates. Eccentricity and non-geodesic past-history
corrections are always sub-leading, indicating that the quasi-circular
approximation applies throughout the transition regime. However, we show that
the OT assumption that the energy and angular momentum evolve linearly with
proper time must be modified in the near-extremal regime. Using our transition
equations, we describe an algorithm to compute the full worldline in proper
time for an extreme mass ratio inspiral (EMRI) and the resultant gravitational
waveform in the high spin limit.
