Magnetic flux emergence from the convection zone into the photosphere and beyond is a critical component of the behavior of large-scale solar magnetism. Flux rarely emerges amid field-free areas at the surface, but when it does, the interaction between the magnetism and plasma flows can be reliably explored. Prior ensemble studies have identified weak flows forming near emergence locations, but the low signal-to-noise ratio (S/N) required averaging over the entire data set, erasing information about variation across the sample. Here, we apply deep learning to achieve an improved S/N, enabling a case-by-case study. We find that these associated flows are dissimilar across instances of emergence and also occur frequently in the quiet convective background. Our analysis suggests the diminished influence of supergranular-scale convective flows and magnetic buoyancy on flux rise. Consistent with numerical evidence, we speculate that small-scale surface turbulence and/or deep convective processes play an outsized role in driving flux emergence.