  Ori and Thorne have discussed the duration and observability (with LISA) of
the transition from circular, equatorial inspiral to plunge for stellar-mass
objects into supermassive ($10^{5}-10^{8}M_{\odot}$) Kerr black holes. We
extend their computation to eccentric Kerr equatorial orbits. Even with orbital
parameters near-exactly determined, we find that there is no universal length
for the transition; rather, the length of the transition depends sensitively --
essentially randomly -- on initial conditions. Still, Ori and Thorne's
zero-eccentricity results are essentially an upper bound on the length of
eccentric transitions involving similar bodies (e.g., $a$ fixed). Hence the
implications for observations are no better: if the massive body is
$M=10^{6}M_{\odot}$, the captured body has mass $m$, and the process occurs at
distance $d$ from LISA, then $S/N \lesssim (m/10
M_{\odot})(1\text{Gpc}/d)\times O(1)$, with the precise constant depending on
the black hole spin. For low-mass bodies ($m \lesssim 7 M_\odot$) for which the
event rate is at least vaguely understood, we expect little chance (probably
[much] less than 10%, depending strongly on the astrophysical assumptions) of
LISA detecting a transition event with $S/N>5$ during its run; however, even a
small infusion of higher-mass bodies or a slight improvement in LISA's noise
curve could potentially produce $S/N>5$ transition events during LISA's
lifetime.
