At four months, distance vision is fully developed and babies can follow moving objects with their eyes. (womenandinfants.org)
Many of these core structures are still found in our brains: the optic tectum, involved in tracking moving objects with the eyes; the amygdala, which helps us respond to fearful situations; parts of the limbic system, which gives us feelings of reward and helps to lay down memories; and the basal ganglia, which control patterns of movements. (newscientist.com)
Newborns are just learning to follow moving objects with their eyes (called visual tracking), so wait for baby to find the object and then move it a bit more. (candokiddo.com)