  We propose a new method for self-injection of high-quality electron bunches
in the plasma wakefield structure in the blowout regime utilizing a "flying
focus" produced by a drive beam with an energy chirp. In a flying focus the
speed of the density centroid of the drive bunch can be superluminal or
subluminal by utilizing the chromatic dependence of the focusing optics. We
first derive the focal velocity and the characteristic length of the focal spot
in terms of the focal length and an energy chirp. We then demonstrate using
multidimensional particle-in-cell simulations that a wake driven by a
superluminally propagating flying focus of an electron beam can generate
GeV-level electron bunches with ultralow normalized slice emittance ($\sim$30
nm rad), high current ($\sim$ 17 kA), low slice energy-spread ($\sim$0.1%) and
therefore high normalized brightness ($>10^{19}$ A/rad$^2$/m$^2$) in a plasma
of density $\sim10^{19}$ cm$^{-3}$. The injection process is highly
controllable and tunable by changing the focal velocity and shaping the drive
beam current. Near-term experiments at FACET II where the capabilities to
generate tens of kA, <10 fs drivers are planned, could potentially produce
beams with brightness near $10^{20}$ A/rad$^2$/m$^2$.
