We introduce a class of branching processes in which the reproduction or lifetime distribution at a given time depends on the total cumulative number of individuals who have been born in the population until that time. We focus on a continuous-time version of these processes, called total-progeny-dependent birth-and-death processes, and study some of their properties through the analysis of their fluid (deterministic) approximation. These properties include the maximum population size, the total progeny size at extinction, the time to reach the maximum population size, and the time until extinction. As the fluid approach does not allow us to approximate the time until extinction directly, we propose several methods to complement this approach. We also use the fluid approach to study the behaviour of the processes as we increase the magnitude of the individual birth rate.