This work addresses the mismatch problem between the distribution of training data (source) and testing data (target), in the challenging context of dysarthric speech recognition. We focus on Speaker Adaptation (SA) in command speech recognition, where data from multiple sources (i.e., multiple speakers) are available. Specifically, we propose an unsupervised Multi-Source Domain Adaptation (MSDA) algorithm based on optimal-transport, called MSDA via Weighted Joint Optimal Transport (MSDA-WJDOT). We achieve a Command Error Rate relative reduction of 16% and 7% over the speaker-independent model and the best competitor method, respectively. The strength of the proposed approach is that, differently from any other existing SA method, it offers an interpretable model that can also be exploited, in this context, to diagnose dysarthria without any specific training. Indeed, it provides a closeness measure between the target and the source speakers, reflecting their similarity in terms of speech characteristics. Based on the similarity between the target speaker and the healthy/dysarthric source speakers, we then define the healthy/dysarthric score of the target speaker that we leverage to perform dysarthria detection. This approach does not require any additional training and achieves a 95% accuracy in the dysarthria diagnosis.