This paper presents a novel system architecture that integrates blind source separation with joint beat and downbeat tracking in musical audio signals. The source separation module segregates the percussive and non-percussive components of the input signal, over which beat and downbeat tracking are performed separately and then the results are aggregated with a learnable fusion mechanism. This way, the system can adaptively determine how much the tracking result for an input signal should depend on the input's percussive or non-percussive components. Evaluation on four testing sets that feature different levels of presence of drum sounds shows that the new architecture consistently outperforms the widely-adopted baseline architecture that does not employ source separation.