Respiration is a critical vital sign for infants, and continuous respiratory monitoring is particularly important for newborns. However, neonates are sensitive and contact-based sensors present challenges in comfort, hygiene, and skin health, especially for preterm babies. As a step toward fully automatic, continuous, and contactless respiratory monitoring, we develop a deep-learning method for estimating respiratory rate and waveform from plain video footage in natural settings. Our automated infant respiration flow-based network (AIRFlowNet) combines video-extracted optical flow input and spatiotemporal convolutional processing tuned to the infant domain. We support our model with the first public annotated infant respiration dataset with 125 videos (AIR-125), drawn from eight infant subjects, set varied pose, lighting, and camera conditions. We include manual respiration annotations and optimize AIRFlowNet training on them using a novel spectral bandpass loss function. When trained and tested on the AIR-125 infant data, our method significantly outperforms other state-of-the-art methods in respiratory rate estimation, achieving a mean absolute error of $\sim$2.9 breaths per minute, compared to $\sim$4.7--6.2 for other public models designed for adult subjects and more uniform environments.