Both the predictive simulations of runaway-electron-induced damage and the development of optimal runaway electron handling strategies require the simulation of the runaway electron distribution in (at least) 3 dimensions: 1D spatial and 2D momentum space. There have been significant advances in non-linear MHD simulations of tokamak disruptions recently, but there is still room for self-consistent kinetic simulations more simplistic handling of the magnetic geometry, like DREAM. Simulations with DREAM have been performed for the parameter range of reactor-scale tokamaks. The simulation starts with a thermal quench and continues throughout the following MHD-inactive period. Our goal was to qualitatively describe the evolution of the runaway electron distribution function, identify the parameters governing the runaway electron dynamics, and relate the numerical results to previously published simplified models of runaway electron distributions. Results show that, provided we can avoid significant hot-tail formation of runaway electrons, the primary generation is dominated by Compton scattering while most runaway electrons are generated consequently by the avalanche mechanism. The high electric field during the thermal quench results in a distribution function monotonic in the energy. On a longer time scale, the distribution can gradually turn into a bump-on-tail type. Dependence of this time evolution is studied for different disruption and modelling parameters.