We investigate the process of accretion of interstellar gas by the Galactic population of brown dwarfs over very long timescales typical for physical eschatology. In particular, we use the classical Hoyle-Lyttleton-Bondi accretion model to investigate the rate at which brown dwarfs collect enough additional mass to become red dwarfs, accretion-induced changes in the mass function of the low- mass objects, and the corresponding accretion heating of brown dwarfs. In addition, we show how we can make the definition of the final mass function for stellar objects more precise.