We study the mass flow rate through a disc resulting from a varying mass-supply rate. Variable mass-supply rate occurs, for example, during disc state transitions, and in interacting eccentric binaries. It is, however, damped by the viscosity of the disc. Here, we calculate this damping in detail. We derive an analytical description of the propagation of the flow rate using the solution of Lynden-Bell & Pringle, in which the disc is assumed to extend to infinity. In particular, we derive the accretion-rate Green's function, and its Fourier transform, which gives the fractional damping at a given variability frequency. We then compare this model to that of a finite disc with the mass supply at its outer edge. We find significant differences with respect to the infinite disc solution, which we find to overestimate the viscous damping. In particular, the asymptotic form of the Green's function is power law for the infinite disc and exponential for the finite one. We then find a simple fitting form for the latter, and also calculate its Fourier transform. In general, the damping becomes very strong when the viscous time at the outer edge of the disc becomes longer than the modulation time-scale. We apply our results to a number of astrophysical systems. We find the effect is much stronger in low-mass X-ray binaries, where the disc size is comparable to that of the Roche lobe, than in high-mass binaries, where the wind-fed disc can have a much smaller size.