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Coronal mass ejections (CMEs) are energetic releases of large‐scale magnetic structures from the Sun. CMEs can have impacts on spacecraft and at Earth. This trajectory is typically assumed to be radial, but often the CME moves outward with some spatial offset from the source region where the eruption initially occurred. A CME is frequently accompanied by a prominence eruption, a movement of cool, dense material up into the corona that can be ejected or fall back down. We investigate eruptions in which some portion of the prominence material falls back to the Sun along field lines which have reconfigured in the eruption, rather than draining back to the source or escaping with the CME. Using a method called persistence mapping, 304 Å images from the Solar Dynamics Observatory (SDO), and coronagraph images from the Solar and Heliospheric Observatory, we measure and compare the offsets in latitude of 20 CMEs and their respective prominences with respect to the source region. The 20 events were chosen to sample over the first 10 years of the SDO mission. We find that the offsets are correlated. We find no difference between eruptions offset toward the equator or the poles, suggesting that the offset is a result of local changes in the eruptive field, rather than of the Sun's global magnetic field structure. These findings help us contextualize individual eruptions and highlight changes in the local magnetic field associated with the prominence eruption. Plain Language Summary: Solar eruptions can have impacts on spacecraft and at Earth. We investigate two components of solar eruptions: the coronal mass ejection (CME), which is an energetic release of large‐scale magnetic structure from the Sun, and partially erupted prominence material (PEPM) that falls back to the Sun along changing magnetic field lines during the eruption. For multiple eruptions, we measure and compare the offset in latitude of both the CME and PEPM from the source region where the eruption originated. We find that the two offsets are correlated, indicating that the changing magnetic field topology impacts both of these components. These findings can help us contextualize individual eruptions and highlight the changing local magnetic field. Key Points: We examine the motion of partially erupted prominence material (PEPM) and coronal mass ejection (CME) relative to eruptive source locationWe find a correlation between the offsets in latitude from the source locationPEPM can be a predictor of CME direction relative to source region and highlight evolving magnetic topology [ABSTRACT FROM AUTHOR]