Objective. In proton pencil beam scanning (PBS) continuous delivery, the beam is continuously delivered without interruptions between spots. For synchrotron-based systems, the extracted beam current exhibits a spill structure, and recent publications on beam current measurements have demonstrated significant fluctuations around the nominal values. These fluctuations potentially lead to dose deviations from those calculated assuming a stable beam current. This study investigated the dosimetric implications of such beam current fluctuations during proton PBS continuous scanning. Approach. Using representative clinical proton PBS plans, we performed simulations to mimic a worst-case clinical delivery environment with beam current varies from 50% to 250% of the nominal values. The simulations used the beam delivery parameters optimized for the best beam delivery efficiency of the upcoming particle therapy system at Mayo Clinic Florida. We reconstructed the simulated delivered dose distributions and evaluated the dosimetric impact of beam current fluctuations. Main results. Despite significant beam current fluctuations resulting in deviations at each spot level, the overall dose distributions were nearly identical to those assuming a stable beam current. The 1 mm/1% Gamma passing rate was 100% for all plans. Less than 0.2% root mean square error was observed in the planning target volume dose-volume histogram. Minimal differences were observed in all dosimetric evaluation metrics. Significance. Our findings demonstrate that with our beam delivery system and clinical planning practice, while significant beam current fluctuations may result in large local move monitor unit deviations at each spot level, the overall impact on the dose distribution is minimal.