Multiple sclerosis (MS) expresses substantial clinical and radiological heterogeneity, which poses significant challenges for automatic lesion segmentation. The current deep learning-based SOTA is highly susceptible to changes in both distribution, e.g., changes in scanner; as well as the structure of inputs, evident in the current divide between cross-sectional and longitudinal approaches. We introduce TimeLesSeg, a unified contrast-agnostic framework designed to segment MS lesions regardless of the presence of a temporal dimension in its inputs, with a single convolutional neural network. Our approach models pathological priors through lesion masks, which are processed together with the current scan. Cross-sectional processing is enabled by exposing the model to training cases where no prior information is available, which are modeled with an empty mask, allowing it to operate seamlessly in both scenarios. To overcome the scarcity and inconsistency of longitudinal datasets, we propose a novel generative pipeline in which patterns of lesion evolution are simulated by stochastically deforming each individual lesion with morphological operations, producing realistic prior timepoints. In parallel, we achieve contrast agnosticism through Gaussian mixture model-based domain randomization, enabling the network to experience a wide spectrum of intensity profiles. Results on three publicly available and two in-house datasets show that TimeLesSeg outperforms the contrast-agnostic state of the art on single-modality inputs across overlap- and distance-based metrics. In longitudinal processing, our method outperforms SAMSEG, and captures lesion load dynamics more accurately than both the former and LST-AI. All source code related to the development of TimeLesSeg is available at https://github.com/NeuroADaS-Lab/TimeLesSeg.