  It is known that massive black holes (MBHs) of $10^{5-7}\,M_\odot$ could
capture small compact objects to form extreme-mass-ratio inspirals (EMRIs).
Such systems emit gravitational waves (GWs) in the band of the Laser
Interferometer Space Antenna (LISA) and are ideal probes of the space-time
geometry of MBHs. Recently, we have shown that MBHs could also capture
stellar-mass binary black holes (about $10\,M_\odot$) to form binary-EMRIs
(b-EMRIs) and, interestingly, a large fraction of the binaries coalesce due to
the tidal perturbation by the MBHs. Here we further show that the coalescence
could be detected by LISA as glitches in EMRI signals. We propose an experiment
to use the multi-band ($10^2$ and $10^{-3}$ Hz) glitch signals to test gravity
theories. Our simulations suggest that the experiment could measure the mass
and linear momentum lost via GW radiation, as well as constrain the mass of
gravitons, to a precision that is one order of magnitude better than the
current limit.
