In everyday acoustic scenes, listeners face the challenge of selectively attending to a sound source and maintaining attention on that source long enough to extract meaning. This task is made more daunting by frequent perceptual discontinuities in the acoustic scene: talkers move in space and conversations switch from one speaker to another in a background of many other sources. The inherent dynamics of such switches directly impact our ability to sustain attention. Here we asked how discontinuity in talker voice affects the ability to focus auditory attention to sounds from a particular location as well as neural correlates of underlying processes. During electroencephalography recordings, listeners attended to a stream of spoken syllables from one direction while ignoring distracting syllables from a different talker from the opposite hemifield. On some trials, the talker switched locations in the middle of the streams, creating a discontinuity. This switch disrupted attentional modulation of cortical responses; specifically, event-related potentials evoked by syllables in the to-be-attended direction were suppressed and power in alpha oscillations (8–12 Hz) were reduced following the discontinuity. Importantly, at an individual level, the ability to maintain attention to a target stream and report its content, despite the discontinuity, correlates with the magnitude of the disruption of these cortical responses. These results have implications for understanding cortical mechanisms supporting attention. The changes in the cortical responses may serve as a predictor of how well individuals can communicate in complex acoustic scenes and may help in the development of assistive devices and interventions to aid clinical populations.