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We explore a novel class of multi-particle dark sectors, called Inelastic Boosted Dark Matter (iBDM). These models are constructed by combining properties of particles that scatter off matter by making transitions to heavier states (Inelastic Dark Matter) with properties of particles that are produced with a large Lorentz boost in annihilation processes in the galactic halo (Boosted Dark Matter). This combination leads to new signals that can be observed at ordinary direct detection experiments, but require unconventional searches for energetic recoil electrons in coincidence with displaced multi-track events. Related experimental strategies can also be used to probe MeV-range boosted dark matter via their interactions with electrons inside the target material.