Manual segmentation of lesions, required for radiotherapy planning and follow-up, is time-consuming and error-prone. Automatic detection and segmentation can assist radiologists in these tasks. This work explores the automated detection and segmentation of brain metastases (BMs) in longitudinal MRIs. It focuses on several important aspects: identifying and segmenting new lesions for screening and treatment planning, re-segmenting lesions in successive images using prior lesion locations as an additional input channel, and performing multi-component segmentation to distinguish between enhancing tissue, edema, and necrosis. The key component of the proposed approach is to propagate the lesion mask from the previous time point to improve the detection performance, which we refer to as "re-segmentation". The retrospective data includes 518 metastases in 184 contrast-enhanced T1-weighted MRIs originating from 49 patients (63% male, 37% female). 131 time-points (36 patients, 418 BMs) are used for cross-validation, the remaining 53 time-points (13 patients, 100 BMs) are used for testing. The lesions were manually delineated with label 1: enhancing lesion, label 2: edema, and label 3: necrosis. One-tailed t-tests are used to compare model performance including multiple segmentation and detection metrics. Significance is considered as p < 0.05. A Dice Similarity Coefficient (DSC) of 0.79 and -score of 0.80 are obtained for the segmentation of new lesions. On follow-up scans, the re-segmentation model significantly outperforms the segmentation model (DSC and 0.78 and 0.88 vs 0.56 and 0.60). The re-segmentation model also significantly outperforms the simple segmentation model on the enhancing lesion (DSC 0.76 vs 0.53) and edema (0.52 vs 0.47) components, while similar scores are obtained on the necrosis component (0.62 vs 0.63). Additionally, we analyze the correlation between lesion size and segmentation performance, as demonstrated in various studies that highlight the challenges in segmenting small lesions. Our findings indicate that this correlation disappears when utilizing the re-segmentation approach and evaluating with the unbiased normalized DSC. In conclusion, the automated segmentation of new lesions and subsequent re-segmentation in follow-up images was achievable, with high level of performance obtained for single- and multiple-component segmentation tasks.