Motion remains a significant challenge in MRI, particularly for body imaging and prolonged multi-contrast acquisitions. Although numerous classical and deep learning approaches have been developed for nonrigid motion correction, many require additional calibration, pre-training, or may have limited generalizability across protocols and subjects. Inspired by implicit neural representations (INR), we explore the application of a Neural Space-Time Model (NSTM) for motion-corrected MRI reconstruction without the need for priors or external training data. In this approach, the image and motion fields are modeled as continuous functions of spatial and temporal coordinates, with separate learnable time encodings used to disentangle contrast dynamics from motion evolution. The two INRs are trained jointly by minimizing a loss formulated directly on the acquired k-space measurements. Preliminary results in motion simulations and cardiac in vivo data show that NSTM recovers plausible nonrigid motion fields and multi-frame images. In in vivo dynamic contrast-enhanced MRI (DCE-MRI), it effectively captures contrast dynamics at a temporal resolution of 1.5 s, highlighting its potential as a tool for robust dynamic MRI reconstruction. The code repository with the full model implementation is available at: https://github.com/nurdinova/nstm_mri_moco