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Accurate brain segmentation is critical for magnetic resonance imaging (MRI) analysis pipelines. Machine-learning-based brain MR image segmentation methods are among the state-of-the-art techniques for this task. Nevertheless, the segmentations produced by machine learning models often degrade in the presence of expected domain shifts between the test and train sets data distributions. These domain shifts are expected due to several factors, such as scanner hardware and software differences, technology updates, and differences in MRI acquisition parameters. Domain adaptation (DA) methods can make machine learning models more resilient to these domain shifts. This paper proposes a benchmark for investigating DA techniques for brain MR image segmentation using data collected across sites with scanners from different vendors (Philips, Siemens, and General Electric). Our work provides labeled data, publicly available source code for a set of baseline and DA models, and a benchmark for assessing different brain MR image segmentation techniques. We applied the proposed benchmark to evaluate two segmentation tasks: skull-stripping; and white-matter, gray-matter, and cerebrospinal fluid segmentation, but the benchmark can be extended to other brain structures. Our main findings during the development of this benchmark are that there is not a single DA technique that consistently outperforms others, and hyperparameter tuning and computational times for these methods still pose a challenge before broader adoption of these methods in the clinical practice.