Autonomous flight of flapping-wing robots is a major challenge for robot perception. Most of the previous senseand-avoid works have studied the problem of obstacle avoidance for flapping-wing robots considering only static obstacles. This paper presents a fully onboard dynamic sense-and-avoid scheme for large-scale ornithopters using event cameras. These sensors trigger pixel information due to changes of illumination in the scene such as those produced by dynamic objects. The method performs event-by-event processing in low-cost hardware such as those onboard small aerial vehicles. The proposed scheme detects obstacles and evaluates possible collisions with the robot body. The onboard controller actuates over the horizontal and vertical tail deflections to execute the avoidance maneuver. The scheme is validated in both indoor and outdoor scenarios using obstacles of different shapes and sizes. To the best of the authors' knowledge, this is the first event-based method for dynamic obstacle avoidance in a flapping-wing robot.