Modeling disturbances in an ecosystem involves choices along many axes, including whether the disturbances occur continuously in time or at discrete time points. Recently, flow-kick maps have emerged as a framework to study impulsive disturbances that occur periodically in time. For example, instead of representing harvests of a logistic population with the ODE dx/dt = x(1 - x) - h, a flow-kick map composes flow for time t governed x(1 - x) with harvests (kicks) that send x to x - th. Iterating the flow-kick map generates discrete dynamics that capture the interplay between disturbance and recovery processes. To support informed use of flow-kick maps as a modeling tool, we explore their connections to classic ODE disturbance models. We find that continuous disturbance models can be recovered as limits of repeated, discrete ones by taking kicks and flow times to zero while maintaining their proportion. Turning this perspective around, we ask what dynamics we lose, keep, or gain by discretizing disturbance. Under suitable nondegeneracy conditions, fixed points and bifurcations for an ODE model continue to flow-kick models that feature similar disturbance rates and sufficiently small flow times. However, larger flow times can destroy these features or introduce new ones. Flow-kick models thus have the potential to reveal novel insights about disturbance dynamics.