In this paper, we investigate the dynamics of muddy-seabed motions induced by a surface solitary wave. The muddy seabed is characterized as Bingham-plastic mud. We focus our attention on the situations where the horizontal scale of the wave-induced mud flow is much larger than the vertical scale. The thickness of the mud layer is also assumed to be much smaller than the water depth above. With these simplifications, the dynamic pressure in the mud column remains a constant and the vertical displacement of the water-mud interface is negligible. The horizontal gradient of the wave-induced dynamic pressure along the water-seabed interface drives the motions in the mud bed. For a Bingham-plastic muddy seafloor, the mud moves either like a solid (plug flow) or like a viscous fluid (shear flow) depending on whether the magnitude of shear stress is in excess of the yield stress. Velocities inside these two different flow regimes and the location(s) of the yield surface(s) vary in time as functions of water-mud interfacial pressure gradient and the properties of the Bingham-plastic mud. A semi-analytical approach is developed in this paper to analyse the motions inside the mud bed under a surface solitary wave loading. Three possible scenarios are discussed to illustrate the complexity of the seafloor responses. The formula for the damping rate caused by the energy dissipation inside the muddy seabed is also derived. Using realistic values of the physical parameters, the present results for damping rate agree qualitatively with the available field observations.