  Extreme-mass-ratio inspirals (EMRIs) are important sources for space-borne
gravitational-wave (GW) detectors. Such a source normally consists of a
stellar-mass black hole (BH) and a Kerr supermassive BH (SMBH), but recent
astrophysical models predict that the small body could also be a stellar-mass
binary BH (BBH). A BBH reaching several gravitational radii of a SMBH will
induce rich observable signatures in the waveform, but the current numerical
tools are insufficient to simulate such a triple system while capturing the
essential relativistic effects. Here we solve the problem by studying the
dynamics in a frame freely falling alongside the BBH. Since the BBH is normally
non-relativistic and much smaller than the curvature radius of the Kerr
background, the evolution in the free-fall frame reduces to essentially
Newtonian dynamics, except for a perturbative gravito-electromagnetic (GEM)
force induced by the curved background. We use this method to study the BBHs on
near-circular orbits around a SMBH and track their evolution down to a distance
of $2-3$ gravitational radii from the SMBH. Our simulations reveal a series of
dynamical effects which are not shown in the previous studies using
conventional methods. The most notable one is a radial oscillation and
azimuthal drift of the BBH relative to the SMBH. These results provide new
insight into the evolution and detection of the EMRIs containing BBHs.
