It is natural to regard most animal movement as a continuous-time process, generally observed at discrete times. Most existing statistical methods for movement data ignore this; the remainder mostly use discrete-time approximations, the statistical properties of which have not been widely studied, or are limited to special cases. We aim to facilitate wider use of continuous-time modelling for realistic problems. We develop novel methodology which allows exact Bayesian statistical analysis for a rich class of movement models with behavioural switching in continuous time, without any need for time discretization error. We represent the times of changes in behaviour as forming a thinned Poisson process, allowing exact simulation and Markov chain Monte Carlo inference. The methodology applies to data that are regular or irregular in time, with or without missing values. We apply these methods to GPS data from two animals, a fisher (Pekania [Martes] pennanti) and a wild boar (Sus scrofa), using models with both spatial and temporal heterogeneity. We are able to identify and describe differences in movement behaviour across habitats and over time. Our methods allow exact fitting of realistically complex movement models, incorporating environmental information. They also provide an essential point of reference for evaluating other existing and future approximate methods for continuous-time inference.