Dialogue technologies such as Amazon's Alexa have the potential to transform the healthcare industry. However, current systems are not yet naturally interactive: they are often turn-based, have naive end-of-turn detection and completely ignore many types of verbal and visual feedback - such as backchannels, hesitation markers, filled pauses, gaze, brow furrows and disfluencies - that are crucial in guiding and managing the conversational process. This is especially important in the healthcare industry as target users of Spoken Dialogue Systems (SDSs) are likely to be frail, older, distracted or suffer from cognitive decline which impacts their ability to make effective use of current systems. In this paper, we outline some of the challenges that are in urgent need of further research, including Incremental Speech Recognition and a systematic study of the interactional patterns in conversation that are potentially diagnostic of dementia, and how these might inform research on and the design of the next generation of SDSs.