The dynamic behavior of disturbances in the vicinity of a pair of magnetically connected three-dimensional null points is examined. The aim is to investigate how nonlinear disturbances lead to strong localized currents that initiate magnetic reconnection at the separator. The problem is formulated in an incompressible cylindrical geometry by superposing arbitrary disturbance fields onto a “background” two-null field. Two different regimes are found for the dynamic evolution, depending on the relative strengths of the background magnetic and velocity fields. In one regime, disturbance pulses split into ingoing and outgoing components, which propagate along the background field lines. In the other “flux pileup” regime, a strong driving flow localizes the disturbances toward the null point pair. Current structures aligned with the spines, fans, and separator present in the field are found to result, and the structure of these currents and their scaling with resistivity is investigated.