Domain-adaptive trajectory imitation is a skill that some predators learn for survival, by mapping dynamic information from one domain (their speed and steering direction) to a different domain (current position of the moving prey). An intelligent agent with this skill could be exploited for a diversity of tasks, including the recognition of abnormal motion in traffic once it has learned to imitate representative trajectories. Towards this direction, we propose DATI, a deep reinforcement learning agent designed for domain-adaptive trajectory imitation using a cycle-consistent generative adversarial method. Our experiments on a variety of synthetic families of reference trajectories show that DATI outperforms baseline methods for imitation learning and optimal control in this setting, keeping the same per-task hyperparameters. Its generalization to a real-world scenario is shown through the discovery of abnormal motion patterns in maritime traffic, opening the door for the use of deep reinforcement learning methods for spatially-unconstrained trajectory data mining.