In real-world machine learning scenarios, training data are frequently weakly annotated and distributionally misaligned with deployment environments. Specifically, label ambiguity may arise when each instance is associated with a set of candidate labels, and distribution shifts between training and testing are common in practice. Although Partial Label Learning (PLL) and Unsupervised Domain Adaptation (UDA) have been extensively studied individually, they frequently co-occur in practice. For instance, in cross-hospital medical image analysis, datasets may exhibit both inconsistent diagnostic labels due to variations in expert interpretation (label ambiguity) and significant differences in imaging equipment or patient demographics (distribution shift). However, Partial-Label Unsupervised Domain Adaptation (PLUDA) has received limited attention as a unified problem. In this paper, a unified generalization bound is established for Partial-Label Unsupervised Domain Adaptation (PLUDA) and three critical limitations causing existing approaches to fail: ambiguity degree, ideal joint error, and model complexity remain uncontrolled. Motivated by these theoretical insights, we propose Dual-Smoothing over Manifold and Parameter (DSMP) to control all three factors. DSMP employs manifold-based representation smoothing via Laplacian smoothing based on adaptive multi-kernel RKHS similarity and candidate set refinement to address the three critical limitations. Moreover, DSMP leverages sharpness-aware parameter smoothing to ensure stable optimization under weak supervision through loss landscape flattening. Extensive experiments demonstrate that DSMP outperforms existing baselines, achieving superior cross-domain generalization from weakly labeled sources. This work provides theoretical insights and a principled solution to the previously underexplored yet practically important PLUDA problem.