This paper extends the Boltzmann hereditary delay integral approach previously used to predict the anelastic response of a floating flexible plate to a steadily moving load, to describe the evolution of the response when the load is impulsively started from rest. Asymptotic analysis demonstrates that the ultimate one-dimensional and two-dimensional deflections (due to a line load and a point load, respectively) are consistent with the results obtained from the earlier time-independent viscoelastic theory. Transients in the two-dimensional case decay much more rapidly than their counterparts in the one-dimensional case. Steady-state deflections are approached at all load speeds in the two-dimensional theory, including the critical load speed (coincident with the minimum phase speed of hybrid flexural–gravity waves) and the gravity wave speed.