To accurately analyze changes of anatomical structures in longitudinal imaging studies, consistent segmentation across multiple time-points is required. Existing solutions often involve independent registration and segmentation components. Registration between time-points is used either as a prior for segmentation in a subsequent time point or to perform segmentation in a common space. In this work, we propose a novel hybrid convolutional neural network (CNN) that integrates segmentation and registration into a single procedure. We hypothesize that the joint optimization leads to increased performance on both tasks. The hybrid CNN is trained by minimizing an integrated loss function composed of four different terms, measuring segmentation accuracy, similarity between registered images, deformation field smoothness, and segmentation consistency. We applied this method to the segmentation of white matter tracts, describing functionally grouped axonal fibers, using N=8045 longitudinal brain MRI data of 3249 individuals. The proposed method was compared with two multistage pipelines using two existing segmentation methods combined with a conventional deformable registration algorithm. In addition, we assessed the added value of the joint optimization for segmentation and registration separately. The hybrid CNN yielded significantly higher accuracy, consistency and reproducibility of segmentation than the multistage pipelines, and was orders of magnitude faster. Therefore, we expect it can serve as a novel tool to support clinical and epidemiological analyses on understanding microstructural brain changes over time.