Ocular position drifts during gaze fixation are significantly less well understood than microsaccades. We recently identified a short-latency ocular position drift response, of ∼1 min arc amplitude, that is triggered within <100 ms by visual onsets. This systematic eye movement response is feature-tuned and seems to be coordinated with a simultaneous resetting of the saccadic system by visual stimuli. However, much remains to be learned about the drift response, especially for designing better-informed neurophysiological experiments unraveling its mechanistic substrates. Here we systematically tested multiple new feature tuning properties of drift responses. Using highly precise eye tracking in three male rhesus macaque monkeys, we found that drift responses still occur for tiny foveal visual stimuli. Moreover, the responses exhibit size tuning, scaling their amplitude (both up and down) as a function of stimulus size, and they also possess a monotonically increasing contrast sensitivity curve. Importantly, short-latency drift responses still occur for small peripheral visual targets, which additionally introduce spatially directed modulations in drift trajectories toward the appearing peripheral stimuli. Drift responses also remain predominantly upward even for stimuli exclusively located in the lower visual field and even when starting gaze position is upward. When we checked the timing of drift responses, we found it was better synchronized to stimulus-induced saccadic inhibition than to stimulus onset. These results, along with a suppression of drift response amplitudes by peristimulus saccades, suggest that drift responses reflect the rapid impacts of short-latency and feature-tuned visual neural activity on final oculomotor control circuitry in the brain.