There are multiple applications where tracking and classification of targets detected in multibeam sonar data is required, ranging from environmental studies to marine security. Target tracking from a fixed frame of reference is well-established, but if the sonar is mounted to a surface-following platform or vessel, the tracking task is complicated by the moving frame of reference. Here, accelerometer data is integrated into a Kalman filter-based tracking scheme to correct for the motion of a non-stationary sonar. This technique is first verified using a vessel-mounted sonar and target with known position. Tracking of marine animals from a moving platform is then demonstrated in real-time for the case of a sonar mounted on the hull of a wave energy converter. These target tracks are classified in real time (e.g., diving bird, seal) using a Random Forest algorithm. The effectiveness and limitations of real-time target tracking and classification from a moving multibeam sonar are discussed.