Computational imaging reconstructions from multiple measurements that are captured sequentially often suffer from motion artifacts if the scene is dynamic. We propose a neural space–time model (NSTM) that jointly estimates the scene and its motion dynamics, without data priors or pre-training. Hence, we can both remove motion artifacts and resolve sample dynamics from the same set of raw measurements used for the conventional reconstruction. We demonstrate NSTM in three computational imaging systems: differential phase-contrast microscopy, three-dimensional structured illumination microscopy and rolling-shutter DiffuserCam. We show that NSTM can recover subcellular motion dynamics and thus reduce the misinterpretation of living systems caused by motion artifacts.