In this paper we explore the evolution of transport capacity on networks with stochastic incidence of damage and accumulation of faults in their connections. For each damaged configuration of the network, we analyze a Markovian random walker that hops over weighted links that quantify the capacity of transport of each connection. The weights of the links in the network evolve due to randomly occurring damage effects that reduce gradually the transport capacity of the structure. We introduce a global measure to determine the functionality of each configuration and how the system ages due to the accumulation of damage that cannot be repaired completely. Then, by assuming a minimum value of the functionality required for the system to be "alive," we explore the statistics of the lifetimes for several realizations of this process in different types of networks. Finally, we analyze the characteristic longevity of such a system and its relation with the "complexity" of the network structure. One finding is that systems with greater complexity live longer. Our approach introduces a model of aging processes relating the reduction of functionality with the accumulation of "misrepairs" and the lifetime of a complex system.