This paper presents a new approach for performing fast multiobjective optimization of the design of moving nonlinear electromagnetic (EM) devices using parametric reduced order EM field models. The approach minimizes the high computational costs that arises from the frequent evaluation of large scale spatially discretized EM field models during the optimization process run. The performance of the proposed approach is demonstrated by carrying out a multiobjective design optimization of an electromagnetic device with a moving component and nonlinear material properties.