The kinetic evolution in velocity space of a minority suprathermal electron population that is undergoing the anomalous Doppler instability (ADI) is investigated using the results from fully nonlinear numerical simulations that self-consistently evolve particles and fields in a plasma. Electron trajectories in phase space during different stages of the ADI are captured, and are analysed in relation to the characteristics of the excited electric fields and of the overall distribution of particles. For some electrons, trapping and mirroring effects are observed during the saturation phase. A relationship between the second order moments of the perpendicular electron distribution function and time is established, and is used to investigate the range of applicability of analytical approximations drawn from classical theory, that involve a quasi-linear wave-driven diffusion operator.