It is commonly believed that the visual motion signals are used by the oculomotor system to predict the future position of a moving target. Yet, in most neurophysiological studies, the oculomotor behavior was recorded in over-trained animals who tracked a target whose spatiotemporal trajectory was learned and memorized. The signals that drive the oculomotor system actually correspond to an estimate of target position expected here-and-now (Fleuriet & Goffart 2012) rather than an estimate of any undefined future position. During the last years, we investigated in three monkeys, the limits of the extrapolation properties in the visuo-oculomotor system. First, we studied saccades launched from straight-ahead toward a target moving horizontally in the upper/lower visual field, with a constant, accelerating or decelerating velocity, and for durations that gradually increased from 50 to 800 ms. The different durations and dynamics were tested on different days (shortest durations tested before). Control saccades toward static targets were recorded before all sessions to quantify the extrapolation. After the tracking became smooth (Bourrelly et al. VSS 2014), we tested how this training affected the oculomotor responses to a transient moving target. A different learning procedure was then tested. After brief exposure, the moving target was occluded for a duration that gradually increased over consecutive days. When the tracking exhibited pursuit-like movements during the occlusion, responses to the transient target were again tested. Contrary to the performance after the first training, the interceptive saccades were followed by a slow eye movement, as if the monkeys were tracking an invisible target. Our work shows the limits of the extrapolation within the visuo-oculomotor system. The signals that encode the spatiotemporal trajectory and drive the on-going tracking behavior result from interpolation mechanisms. Our research should allow identifying the neural channels by which the mnemonic signals dynamically guide the on-going tracking response.