We analyze the dynamics of flat-histogram Monte Carlo algorithms that determine the density of states as a function of potential energy. In contrast to conventional Boltzmann sampling, these methods aid equilibration by enforcing a flat energy distribution, which enables a system to escape local energy minima. For two model fluids, we perform an extensive characterization of flat-histogram dynamics, that is, the dynamical behavior when the sampling is such that the system makes a random walk in energy. We show that the tunneling time, defined as the average number of steps required for the system to move between its low- and high-energy bounds, correlates uniquely with the entropy range of the energy window sampled, independent of system size and of the particular system investigated. We also demonstrate that the rate of statistical improvement of flat-histogram calculations scales with the tunneling time, and hence we propose that an optimal workload distribution in parallel implementations of these methods should divide the total entropy range equally among each processor.