Partial label learning (PLL) tackles the problem where each instance is associated with a set of candidate labels, only one of which is the ground-truth label. Most existing PLL approaches assume that both the training and test sets share an identical data distribution. However, this assumption does not hold in many real-world scenarios where the training and test data come from different distributions. In this paper, we formalize this learning scenario as a new problem called partial label unsupervised domain adaptation (PLUDA). To address this challenging PLUDA problem, we propose a novel Prototype Alignment based PLUDA method named PAPLUDA, which dynamically refines the pseudo-labels of instances from both the source and target domains by consulting the outputs of a teacher-student model in a moving-average manner, and bridges the cross-domain discrepancy through inter-domain class-prototype alignment. In addition, a teacher-student model based contrastive regularization is deployed to enhance prediction stability and hence improve the class-prototypes in both domains for PLUDA. Comprehensive experimental results demonstrate that PAPLUDA achieves state-of-the-art performance on the widely used benchmark datasets.