An object moving in the visual field triggers a saccade that brings its image onto the fovea. It is followed by a combination of slow eye movements and catch-up saccades that try to keep the target image on the fovea as long as possible. The accuracy of this ability to track the "here-and-now" location of a visual target contrasts with the spatiotemporally distributed nature of its encoding in the brain. We show in six experimentally naive monkeys how this performance is acquired and gradually evolves during successive daily sessions. During the early exposure, the tracking is mostly saltatory, made of relatively large saccades separated by low eye velocity episodes, demonstrating that accurate (here and now) pursuit is not spontaneous and that gaze direction lags behind its location most of the time. Over the sessions, while the pursuit velocity is enhanced, the gaze is more frequently directed toward the current target location as a consequence of a 25% reduction in the number of catch-up saccades and a 37% reduction in size (for the first saccade). This smoothing is observed at several scales: during the course of single trials, across the set of trials within a session, and over successive sessions. We explain the neurophysiological processes responsible for this combined evolution of saccades and pursuit in the absence of stringent training constraints. More generally, our study shows that the oculomotor system can be used to discover the neural mechanisms underlying the ability to synchronize a motor effector with a dynamic external event.