The time-dependent response of a floating flexible plate to an impulsively started steadily moving load defines the time taken to approach a steady-state deflection due to the load, or indeed whether such a steady state is achieved at all. The asymptotic analysis for large time reported here, for both a concentrated point load and a uniformly distributed circular load, confirms that a steady-state deflection is achieved at both subcritical and supercritical load speeds. This analysis also predicts a logarithmically growing response near the critical speed corresponding to the minimum phase speed of the hybrid waves generated, but an eventual steady-state response when the load speed moves at the shallow water wave speed. These results are supported by numerical computation.