For an infinite system of particles arriving in and departing from a habitat $X$ -- a locally compact Polish space with a positive Radon measure $χ$ -- a Markov process is constructed in an explicit way. Along with its location $x\in X$, each particle is characterized by age $α\geq 0$ -- time since arriving. As the state space one takes the set of marked configurations $\widehatΓ$, equipped with a metric that makes it a complete and separable metric space. The stochastic evolution of the system is described by a Kolmogorov operator $L$, expressed through the measure $χ$ and a departure rate $m(x,α)\geq 0$, and acting on bounded continuous functions $F:\widehatΓ\to \mathds{R}$. For this operator, we pose the martingale problem and show that it has a unique solution, explicitly constructed in the paper. We also prove that the corresponding process has a unique stationary state and is temporarily egrodic if the rate of departure is separated away from zero.