A specific mortality rate process governed by the non-homogeneous Poisson process of point events is considered and its properties are studied. This process can describe the damage accumulation in organisms experiencing external shocks and define its survival characteristics. It is shown that, although the sample paths of the unconditional mortality rate process are monotonically increasing, the population mortality rate can decrease with age and, under certain assumptions, even tend to zero. The corresponding analysis is the main objective of this paper and it is performed using the derived conditional distributions of relevant random parameters. Several meaningful examples are presented and discussed.