Tidal stripping of dark matter from subhalos falling into the Milky Way produces narrow, cold tidal streams as well as more spatially extended ``debris flows'' in the form of shells, sheets, and plumes.Here we focus on the debris flow in the Via Lactea II simulation, and show that this incompletely phase-mixed material exhibits distinctive high velocity behavior. Unlike tidal streams, which may not necessarily intersect the Earth's location, debris flow is spatially uniform at 8 kpc and thus guaranteed to be present in the dark matter flux incident on direct detection experiments. At Earth-frame speeds greater than 450 km/s, debris flow comprises more than half of the dark matter at the Sun's location, and up to 80% at even higher speeds. Therefore, debris flow is most important for experiments that are particularly sensitive to the high speed tail of the dark matter distribution, such as searches for light or inelastic dark matter or experiments with directional sensitivity. We show that debris flow yields a distinctive recoil energy spectrum and a broadening of the distribution of incidence direction.