Multilingual domain adaptation (ML-DA) is widely used to learn new domain knowledge across languages into large language models (LLMs). Although many methods have been proposed to improve domain adaptation, the mechanisms of multilingual knowledge acquisition, how domain knowledge is learned within a language and transferred across languages, remain underexplored. This gap leads to suboptimal performance, particularly in low-resource settings. This work examines the learning dynamics of LLMs during ML-DA. Because prior ML-DA studies often train and evaluate on datasets with mismatched knowledge coverage, we propose AdaXEval, an adaptive evaluation method that builds multiple-choice QA datasets from the same bilingual domain corpus used for training, thereby directly studying multilingual knowledge acquisition. Through continual training of LLMs with diverse data recipes, we track how LLMs acquire domain facts and pinpoint the mechanism behind the transformation process from domain training data to knowledge. Our experiments on a 13B English-Japanese bilingual LLM reveal that cross-lingual transfer remains challenging despite a high-quality bilingual corpus. The code has been released.