Multi-Source cross-lingual transfer learning deals with the transfer of task knowledge from multiple labelled source languages to an unlabeled target language under the language shift. Existing methods typically focus on weighting the predictions produced by language-specific classifiers of different sources that follow a shared encoder. However, all source languages share the same encoder, which is updated by all these languages. The extracted representations inevitably contain different source languages' information, which may disturb the learning of the language-specific classifiers. Additionally, due to the language gap, language-specific classifiers trained with source labels are unable to make accurate predictions for the target language. Both facts impair the model's performance. To address these challenges, we propose a Disentangled and Adaptive Network (DA-Net). Firstly, we devise a feedback-guided collaborative disentanglement method that seeks to purify input representations of classifiers, thereby mitigating mutual interference from multiple sources. Secondly, we propose a class-aware parallel adaptation method that aligns class-level distributions for each source-target language pair, thereby alleviating the language pairs' language gap. Experimental results on three different tasks involving 38 languages validate the effectiveness of our approach.