Sluggishness of sensory processing in the brain leads to lagging of the sensory perception with respect to external events. In the case of a moving sound source localization the lag should be revealed in perception of a moving target behind its actual location. In spite of this lag human subjects often perceive the moving target with forward displacement clearly ahead of its actual position. A forward displacement indicates capability of brain to overcome the sluggishness in its activity and could be considered as a manifestation of sensory anticipation. In this view auditory motion localization can be interpreted as a result of competitive influences of the sluggishness, leading to backward displacement of the perceived position of a moving auditory target, and of the anticipatory effect compensated for this displacement. Both influences can be estimated based on perceived positions of the starting and ending points of a moving target trajectory. The sluggishness starting at the stimulus onset mainly affects the magnitude of the perceived shifts of the starting point. The anticipatory effect based on motion history develops with some delay and affects position judgments at the ending point of the moving target trajectory. Characteristics of subjective shifts of the ending motion point have shown that the relative weights of the sluggishness and anticipatory effects depend on velocity and duration of target motion. Under short-term movement (100-200ms) the sluggishness dominates at high stimulus velocity resulting in underestimation of the ending position of the moving target. At low velocity the perceived end point shifts to the opposite direction and mislocates ahead of the target position. It is suggested that the anticipatory effect could be determined by dynamic shifts in neural response area in the direction opposite to the direction of motion. Properties of the response area shifts occurred in the auditory neurons (inferior colliculus, auditory cortex) and outside of the auditory system (superior colliculus) are considered.