  Extreme mass ratio inspirals (EMRIs) are compact binary systems characterized
by a mass-ratio $q=m/M$ in the range $~10^{-9}-10^{-4}$ and represent primary
gravitational wave (GW) sources for the forthcoming Laser Interferometer Space
Antenna (LISA). While their standard formation channel involves relaxation
processes deflecting compact objects on very low angular momentum orbits around
the central massive black hole, a number of alternative formation channels has
been proposed, including binary tidal break-up, migration in accretion disks
and secular and chaotic dynamics around a massive black hole binary (MBHB). In
this work, we take an extensive closer look at this latter scenario,
investigating how EMRIs can be triggered by a MBHBs, formed in the aftermath of
galaxy mergers. By employing a suite of relativistic three-body simulations, we
evaluate the efficiency of EMRI formation for different parameters of the MBHB,
assessing the importance of both secular and chaotic dynamics. By modelling the
distribution of compact objects in galaxy nuclei, we estimate the resulting
EMRI formation rate, finding that EMRI are produced in a sharp burst, with peak
rates that are 10-100 times higher than the standard two-body relaxation
channel, lasting for 10$^6$--10$^8$ years. By coupling our results with an
estimate of the cosmic MBHB merger rate, we finally forecast that LISA could
observe ${\cal O}(10)$ EMRIs per year formed by this channel.
