The neural processing of auditory motion information shows a pronounced interhemispheric asymmetry. In previous electrophysiological studies, the so-called motion-onset response (MOR), a prominent auditory-evoked potential to the onset of sound motion, was stronger over the hemisphere contralateral to the side of motion. Here, effects of lateral-onset position and direction of motion on MOR contralaterality were investigated. Eighteen listeners were presented with free-field sound stimuli that, after an initial stationary phase at a lateral spatial position within the left or right hemifield, started to move either left- or rightward. The early part of the MOR, the so-called change-N1, exhibited contralaterality that depended on the lateral motion-onset position with stronger activations over the hemisphere contralateral to the side of motion onset, whereas the contralaterality of the later part of the MOR, the so-called change-P2, merely depended on the direction of motion. Cortical source localization indicated that this pattern of contralaterality primarily resulted from asymmetric activation in primary auditory cortex and insula. These findings suggest that the early and late parts of the MOR reflect different phases in auditory motion perception, supporting the notion of a modular organization of discrete processing stages.