Assessing lesions and tracking their progression over time in brain magnetic resonance (MR) images is essential for diagnosing and monitoring multiple sclerosis (MS). Machine learning models have shown promise in automating the segmentation of MS lesions. However, training these models typically requires large, well-annotated datasets. Unfortunately, MS imaging datasets are often limited in size, spread across multiple hospital sites, and exhibit different formats (such as cross-sectional or longitudinal) and annotation styles. This data diversity presents a significant obstacle to developing a unified model for MS lesion segmentation. To address this issue, we introduce SegHeD+, a novel segmentation model that can handle multiple datasets and tasks, accommodating heterogeneous input data and performing segmentation for all lesions, new lesions, and vanishing lesions. We integrate domain knowledge about MS lesions by incorporating longitudinal, anatomical, and volumetric constraints into the segmentation model. Additionally, we perform lesion-level data augmentation to enlarge the training set and further improve segmentation performance. SegHeD+ is evaluated on five MS datasets and demonstrates superior performance in segmenting all, new, and vanishing lesions, surpassing several state-of-the-art methods in the field.