The follow-up of the development of the premature baby is a major component of its clinical care since it has been shown that it can reveal a pathology. However, no method allowing an automated and continuous monitoring of this development has been proposed. Within the framework of the Digi-NewB European project, our team wishes to offer new clinical indices qualifying the maturation of newborns. In this study, we propose a new method to characterize motor activity from video recordings. For this purpose, we have chosen to characterize the motion temporal organization by drawing inspiration from sleep organization. Thus, we propose a fully automatic process allowing to extract motion features and to combine them to estimate a functional age. By investigating two datasets, one of 28.5 hours (manually annotated) from 33 newborns and one of 4,920 hours from 46 newborns, we show that the proposed approach is relevant for monitoring in clinical routine and that the extracted features reflect the maturation of preterm newborns. Indeed, a compact and interpretable model using gestational age and three motion features (mean duration of intervals with motion, total percentage of time spent in motion and number of intervals without motion) was designed to predict post-menstrual age of newborns and showed an admissible mean absolute error of 1.3 weeks. While the temporal organization of motion was not studied clinically due to a lack of technological means, these results open the door to new developments, new investigations and new knowledge on the evolution of motion in newborns.