  Definite descriptions, such as 'the General Chair of KR 2024', are a
semantically transparent device for object identification in knowledge
representation. In first-order modal logic, definite descriptions have been
widely investigated for their non-rigidity, which allows them to designate
different objects (or none at all) at different states. We propose expressive
modal description logics with non-rigid definite descriptions and names, and
investigate decidability and complexity of the satisfaction problem. We first
systematically link satisfiability for the one-variable fragment of first-order
modal logic with counting to our modal description logics. Then, we prove a
promising NEXPTIME-completeness result for concept satisfiability for the
fundamental epistemic multi-agent logic $\mathbf{S5}^{n}$ and its neighbours,
and show that some expressive logics that are undecidable with constant domain
become decidable (but Ackermann-hard) with expanding domains. Finally, we
conduct a fine-grained analysis of decidability of temporal logics.
