Deep learning algorithms promise to improve clinician workflows and patient outcomes. However, these gains have yet to be fully demonstrated in real world clinical settings. In this paper, we describe a human-centered study of a deep learning system used in clinics for the detection of diabetic eye disease. From interviews and observation across eleven clinics in Thailand, we characterize current eye-screening workflows, user expectations for an AI-assisted screening process, and post-deployment experiences. Our findings indicate that several socio-environmental factors impact model performance, nursing workflows, and the patient experience. We draw on these findings to reflect on the value of conducting human-centered evaluative research alongside prospective evaluations of model accuracy.