Dysarthric speech detection (DSD) systems aim to detect characteristics of the neuromotor disorder from speech. Such systems are particularly susceptible to domain mismatch where the training and testing data come from the source and target domains respectively, but the two domains may differ in terms of speech stimuli, disease etiology, etc. It is hard to acquire labelled data in the target domain, due to high costs of annotating sizeable datasets. This paper makes a first attempt to formulate cross-domain DSD as an unsupervised domain adaptation (UDA) problem. We use labelled source-domain data and unlabelled target-domain data, and propose a multi-task learning strategy, including dysarthria presence classification (DPC), domain adversarial training (DAT) and mutual information minimization (MIM), which aim to learn dysarthria-discriminative and domain-invariant biomarker embeddings. Specifically, DPC helps biomarker embeddings capture critical indicators of dysarthria; DAT forces biomarker embeddings to be indistinguishable in source and target domains; and MIM further reduces the correlation between biomarker embeddings and domain-related cues. By treating the UASPEECH and TORGO corpora respectively as the source and target domains, experiments show that the incorporation of UDA attains absolute increases of 22.2% and 20.0% respectively in utterance-level weighted average recall and speaker-level accuracy.