Evolutionary search via the quality-diversity (QD) paradigm can discover highly performing solutions in different behavioural niches, showing considerable potential in complex real-world scenarios such as evolutionary robotics. Yet most QD methods only tackle static tasks that are fixed over time, which is rarely the case in the real world. Unlike noisy environments, where the fitness of an individual changes slightly at every evaluation, dynamic environments simulate tasks where external factors at unknown and irregular intervals alter the performance of the individual with a severity that is unknown a priori. Literature on optimisation in dynamic environments is extensive, yet such environments have not been explored in the context of QD search. This paper introduces a novel and generalisable Dynamic QD methodology that aims to keep the archive of past solutions updated in the case of environment changes. Secondly, we present a novel characterisation of dynamic environments that can be easily applied to well-known benchmarks, with minor interventions to move them from a static task to a dynamic one. Our Dynamic QD intervention is applied on MAP-Elites and CMA-ME, two powerful QD algorithms, and we test the dynamic variants on different dynamic tasks.