Response time data collected from cognitive tasks are a cornerstone of psychology and neuroscience research, yet existing models of these data either make strong assumptions about the data-generating process or are limited to modelling single trials. We introduce task-DyVA, a deep learning framework in which expressive dynamical systems are trained to reproduce sequences of response times observed in data from individual human subjects. Models fitted to a large task-switching dataset captured subject-specific behavioural differences with high temporal precision, including task-switching costs. Through perturbation experiments and analyses of the models' latent dynamics, we find support for a rational account of switch costs in terms of a stability-flexibility trade-off. Thus, our framework can be used to discover interpretable cognitive theories that explain how the brain dynamically gives rise to behaviour.