In this work, we study the influence of orbital viscosity on the evolution of the order-parameter and texture in the B phase of superfluid 3He near a moving boundary. From the redistribution of thermal quasiparticles within the texture, we develop a model which confers a substantial effective mass on the interface, and provides a new mechanism for friction as the boundary moves. We have tested the model against existing data for the motion of an A-B interface whose motion was controlled by a magnetic field. The model allows us to make predictions for the behaviour in experimental situations which involve texture rearrangement arising from motion of the B-phase boundary.