When a moving object abruptly disappears, this profoundly influences its localization by the visual system. In Experiment 1, 2 aligned objects moved across the screen, and 1 of them abruptly disappeared. Observers reported seeing the objects misaligned at the time of the offset, with the continuing object leading. Experiment 2 showed that the perceived forward displacement of the moving object depended on speed and that offsets were localized accurately. Two competing representations of position for moving objects are proposed: 1 based on a spatially extrapolated internal model, and the other based on transient signals elicited by sudden changes in the object trajectory that can correct the forward-shifted position. Experiment 3 measured forward displacements for moving objects that disappeared only for a short time or abruptly reduced contrast by various amounts. Manipulating the relative strength of the 2 position representations in this way resulted in intermediate positions being perceived, with weaker motion signals or stronger transients leading to less forward displacement. This 2-process mechanism is advantageous because it uses available information about object position to maximally reduce spatio-temporal localization errors.