In many real-world applications, the mismatch between distributions of training data (source) and test data (target) significantly degrades the performance of machine learning algorithms. In speech data, causes of this mismatch include different acoustic environments or speaker characteristics. In this paper, we address this issue in the challenging context of dysarthric speech, by multi-source domain/speaker adaptation (MSDA/MSSA). Specifically, we propose the use of an optimal-transport based approach, called MSDA via Weighted Joint Optimal Transport (MSDA-WDJOT). We confront the mismatch problem in dysarthria detection for which the proposed approach outperforms both the Baseline and the state-of-the-art MSDA models, improving the detection accuracy of 0.9% over the best competitor method. We then employ MSDA-WJDOT for dysarthric speaker adaptation in command speech recognition. This provides a Command Error Rate relative reduction of 16% and 7% over the baseline and the best competitor model, respectively. Interestingly, MSDA-WJDOT provides a similarity score between the source and the target, i.e. between speakers in this case. We leverage this similarity measure to define a Dysarthric and Healthy score of the target speaker and diagnose the dysarthria with an accuracy of 95%.