This paper proposes an intersection-movement-based variational inequality formulation for the multi-class dynamic traffic assignment (DTA) problem involving physical queues using the concept of approach proportion. An extragradient method that requires only pseudomonotonicity and Lipschitz continuity for convergence is developed to solve the problem. We also present a car–truck interaction paradox, which states that allowing trucks to travel or increasing the truck flow in a network can improve network performance for cars in terms of the total car travel time. Numerical examples are set up to illustrate the importance of considering multiple vehicle types and their interactions in a DTA model, the effects of various parameters on the occurrence of the paradox, and the performance of the solution algorithm.