The “two-process model” is a promising technique for interpreting stellar chemical abundance data from large-scale surveys (e.g., the Sloan Digital Sky Survey IV/V and the Galactic Archeology with HERMES survey), enabling more quantitative empirical studies of differences in chemical enrichment history between galaxies without relying on detailed yield and evolution models. In this work, we fit two-process model parameters to (1) a luminous giant Milky Way (MW) sample and (2) stars comprising the Sagittarius dwarf galaxy (Sgr). We then use these two sets of model parameters to predict the abundances of 14 elements of stars belonging to the MW and in five of its massive satellite galaxies, analyzing the residuals between the predicted and observed abundances. We find that the model fit to (1) results in large residuals (0.1–0.3 dex) for most metallicity-dependent elements in the metal-rich ([Mg/H] > −0.8) stars of the satellite galaxies. However, the model fit to (2) results in small or no residuals for all elements across all satellite galaxies. Therefore, despite the wide variation in [X/Mg]–[Mg/H] abundance patterns of the satellite galaxies, the two-process framework provides an accurate characterization of their abundance patterns across many elements, but these multielement patterns are systematically different between the dwarf galaxy satellites and the MW disks. We consider a variety of scenarios for the origin of this difference, highlighting the possibility that a large inflow of pristine gas to the MW disk diluted the metallicity of star-forming gas without changing abundance ratios.