We present the flexdiam dialogue management architecture, which was developed in a series of projects dedicated to tailoring spoken interaction to the needs of users with cognitive impairments in an everyday assistive domain, using a multimodal front-end. This hybrid DM architecture affords incremental processing of uncertain input, a flexible, mixed-initiative information grounding process that can be adapted to users’ cognitive capacities and interactive idiosyncrasies, and generic mechanisms that foster transitions in the joint discourse state that are understandable and controllable by those users, in order to effect a robust interaction for users with varying capacities.