Both sediment transport dynamics and the population level of a buffer in automated production line systems can be described by the same class of stochastic differential equations. The ubiquitous noise is generated by continuous-time Markov chains. The probability densities which describe the dynamics are governed by high-order hyperbolic systems of partial differential equations. While this hyperbolic nature clearly exhibits a nondiffusive character of the processes (diffusion would imply a parabolic evolution of the probability densities), we nevertheless can use a central limit theorem which holds for large-time regimes. This enables analytical estimations of the time evolution of the moments of these processes. Particular emphasis is devoted to non-Markovian, dichotomous alternating renewal processes, which enter directly into the description of the applications presented.