A Lagrangian flow model is used to investigate highly nonlinear, dispersive waves generated by moving seabed deformation (MSD) of an otherwise horizontal seabed. Applications include free surface wave responses to horizontal co-seismic displacements and to novel bed-driven wave making systems used in surfing competitions. This paper considers gravity waves in viscous liquid, without restrictions on wave steepness, dispersion coefficient, and flow regime. Numerical computations are carried out using a Moving Particle Explicit method, which provides a Lagrangian flow description with far fewer particles than existing meshless methods. We show that the MSD speed has different effects in shallow and intermediate water depths. In shallow water, raising the MSD speed to a transcritical value promotes generation of leading solitary waves as expected. In supercritical flow, the highly nonlinear dynamics promotes breaking of the precursor soliton. In intermediate depth, wave dynamics is dominated by nonlinearity and dispersion, which act concurrently to generate a large leading wave that travels faster than predicted by linear theory, followed by a train of dispersive, short, steep waves. These waves break, even at subcritical values of MSD speed. We show that strongly nonlinear viscous dynamics occurs in the presence of a steep seabed deformation. This triggers flow separation, linked to strong amplification of wave steepness. Finally, we show that an oscillating MSD is capable of generating higher harmonics by means of nonlinear wave–wave interactions. The model is validated and verified by comparison to previously published experimental data and approximate analytical solutions.