While the population of main sequence debris discs is well constrained, little is known about debris discs around evolved stars. This paper provides a theoretical framework considering the effects of stellar evolution on debris discs, particularly the production and loss of dust within them. Here we repeat a steady state model fit to disc evolution statistics for main sequence A stars, this time using realistic grain optical properties, then evolve that population to consider its detectability at later epochs. Our model predicts that debris discs around giant stars are harder to detect than on the main sequence because radiation pressure is more effective at removing small dust around higher luminosity stars. Just 12% of first ascent giants within 100pc are predicted to have discs detectable with Herschel at 160um. However this is subject to the uncertain effect of sublimation on the disc, which we propose can thus be constrained with such observations. Our model also finds that the rapid decline in stellar luminosity results in only very young white dwarfs having luminous discs. As such systems are on average at larger distances they are hard to detect, but we predict that the stellar parameters most likely to yield a disc detection are a white dwarf at 200pc with cooling age of 0.1Myr, in line with observations of the Helix Nebula. Our model does not predict close-in (<0.01AU) dust, as observed for some white dwarfs, however we find that stellar wind drag leaves significant mass (~10^{-2}Msolar), in bodies up to ~10m in diameter, inside the disc at the end of the AGB phase which may replenish these discs.