Compressed sensing (CS) has been demonstrated to accelerate MRI acquisitions by reconstructing sparse images of good quality from highly undersampled data. Motion during MR scans can cause inconsistencies in k-space data, resulting in strong motion artifacts in the reconstructed images. For CS to be useful in these applications, motion correction techniques need to be combined with the undersampled reconstruction. Recently, joint motion correction and CS approaches have been proposed to partially correct for effects of motion. However, the main limitation of these approaches is that they can only correct for affine deformations. In this work, we propose a novel motion corrected CS framework for free-breathing dynamic cardiac MRI that incorporates a general motion correction formulation directly into the CS reconstruction. This framework can correct for arbitrary affine or nonrigid motion in the CS reconstructed cardiac images, while simultaneously benefiting from highly accelerated MR acquisition. The application of this approach is demonstrated both in simulations and in vivo data for 2D respiratory self-gated free-breathing cardiac CINE MRI, using a golden angle radial acquisition. Results show that this approach allows for the reconstruction of respiratory motion corrected cardiac CINE images with similar quality to breath-held acquisitions.