With its extreme beam parameters, the FACET II facility enables the test of the down ramp injection scheme in the laboratory for the beam-driven plasma wakefield accelerator approach. In addition to the ideal cases studied in previous theoretical work, we investigate the effect of fluctuations in the down ramp plasma source profile on the emittance and current profile of the self-injected beam using 2D and 3D particle-in-cell simulations. We show that down ramps with a length of ∼10  c/ωp can be reproducibly created by generating a shock in a supersonic flow. Simulations show that the emittance of the injected beam using such down ramps is ∼0.1  μm. A gentler ramp can further reduce the slice emittance of the beam to ∼0.03  μm. The emittance of the injected beam depends on the ramp length but is insensitive to the shape of the ramp. However, the current profile of the injected beam can be manipulated by changing the ramp shape. A simulation shows that, when a noncylindrical driver is used, the emittance of the injected beam differs in the two orthogonal transverse planes and is a few times larger than that of the cylindrical driver case. A full-scale simulation where we use a realistic density profile that includes the up ramp, the injecting down ramp, and the exiting down ramp is presented to show that the beam emittance is preserved during the acceleration and extraction of the beam.