  We study mass flow rate through a disc resulting from a varying mass supply
rate. Variable mass supply rate occurs, e.g., 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.
