This article describes a general framework for multiple coil MRI reconstruction in the presence of elastic physiological motion. On the assumption that motion is known or can be predicted, it is shown that the reconstruction problem is equivalent to solving an integral equation--known in the literature as a Fredholm equation of the first kind--with a generalized kernel comprising Fourier and coil sensitivity encoding, modified by physiological motion information. Numerical solutions are found using an iterative linear system solver. The different steps in the numerical resolution are discussed, in particular it is shown how over-determination can be used to improve the conditioning of the generalized encoding operator. Practical implementation requires prior knowledge of displacement fields, so a model of patient motion is described which allows elastic displacements to be predicted from various input signals (e.g., respiratory belts, ECG, navigator echoes), after a free-breathing calibration scan. Practical implementation was demonstrated with a moving phantom setup and in two free-breathing healthy subjects, with images from the thoracic-abdominal region. Results show that the method effectively suppresses the motion blurring/ghosting artifacts, and that scan repetitions can be used as a source of over-determination to improve the reconstruction.