Extracting calcium traces from the neurons of mobile animals is a critical step in the study of the large-scale neuronal dynamics that govern behavior. Accurate activity extraction requires the correction of motion and movement-induced deformations as well as demixing of signals that may overlap spatially due to limitations in optical resolution. Traditionally, non-negative matrix factorization (NMF) methods have been successful in demixing and denoising cellular calcium activity in relatively motionless or pre-registered videos. However, standard NMF methods fail in mobile animals undergoing significant non-rigid motion; similarly, standard image registration methods based on template matching can fail when large changes in activity lead to mismatches with the image template. To address these issues simultaneously, we introduce a deformable non-negative matrix factorization (dNMF) framework that jointly optimizes registration with signal demixing. On simulated data and real C. elegans microscopy videos, dNMF outperforms traditional demixing methods that account for motion and demixing separately. Finally, following the extraction of neural traces from multiple imaging experiments, we develop a quantile regression time-series normalization technique to account for varying neural signal intensity baselines across different animals or different imaging setups.