A key strategy for sub-GeV dark matter direct detection is searches for small ionization signals that arise from dark matter-electron scattering or from the "Migdal" effect in dark matter-nucleus scattering. We show that the theoretical description of both processes is closely related, allowing for a principal mapping between them. We explore this for noble-liquid targets and, for the first time, estimate the Migdal effect in semiconductors using a crystal form factor. We present new constraints using XENON10, XENON100, and SENSEI data, and give projections for proposed experiments.