We discuss the question of the construction of a linear semigroup for the time evolution of the single-event probabilities of general non-Markov processes. It is shown that such a linear semigroup may not exist for all finite times. The consequences are sketched for the description of equilibrium and nonequilibrium systems. Further, the relationship with nonstationary Markov processes is investigated, and some confusion in recent works is cleared up using the example of free Brownian motion.