We study non-Markovian stochastic differential equations with additive noise characterized by a Poisson point process with arbitrary pulse shapes and exponentially distributed intensities. Specifically, analytic results concerning transitions between different correlation regimes and the long-time asymptotic probability distribution functions are shown to be controlled by the shape of the pulses and dissipative parameter as time progresses. This program is motivated by the study of stochastic partial differential equations perturbed by space Poisson disorder and becomes the main focus of applications of the present exact functional approach.