We investigated how processing benefits for direct face/gaze and sudden onset motion depend on stimulus presentation location, specifically eccentricity from fixation. Participants responded to targets that were presented on one of four stimuli that displayed a direct or averted face and gaze either statically or suddenly. Between participants, stimuli were presented at different eccentricities relative to central fixation, spanning 3.3°, 4.3°, 5.5° or 6.5° of the visual field. Replicating previous studies, we found processing advantages for direct (vs. averted) face/gaze and motion onset (vs. static stimuli). Critically, while the motion-onset advantage increased with increasing distance to the center, the face/gaze direction advantage was not significantly modulated by target eccentricity. Results from a control experiment with eye tracking indicate that face/gaze direction could be accurately discriminated even at the largest eccentricity. These findings demonstrate a distinction between the processing of basic facial and gaze signals and exogenous motion cues, which may be based on functional differences between central and peripheral retinal regions. Moreover, the results highlight the importance of taking specific stimulus properties into account when studying perception and attention in the periphery.