Plates and the convective mantle interact with each other over geological time scales, leading to mantle flow, plate motion, and deformation along plate boundaries.  At convergent boundaries undergoing subduction, the role played by mantle drag remains poorly understood despite its potential impact on subduction dynamics, and in turn on the deformation regime of the overriding plate. Previous studies were generally conducted in two dimensions, limiting their ability to faithfully reproduce processes taking place on Earth. Instead, in this study, we present 11 three-dimensional analog models of subduction at the scale of the upper mantle, including an overriding plate, and in which we control mantle drag at the base of the lower or upper plate by imposing a controlled unidirectional background mantle flow perpendicular to the trench. We systematically vary the velocity and the direction of the imposed horizontal mantle flow and quantify its impact on horizontal and vertical upper plate deformations, plate and subduction velocities, and the geometry of the slab. The geometry of the slab is only marginally affected by the velocity and direction of the mantle flow. In the absence of mantle flow, slab rolls back and deformation is accommodated by trench-orthogonal stretching in the upper plate. Instead, the addition of a background flow dragging the lower or upper plate toward the trench  systematically results either in the absence of upper plate deformation, or in trench-orthogonal shortening with strain rates that increase linearly with increasing mantle flow. We show that the upper plate strain rate is primarily controlled by the velocity of the free plate in the model, which itself results from the drag exerted by the mantle at the base of the plate. Coupling between mantle and plate is larger for models with flow directed toward the upper plate, resulting in strain rates that are about three times larger than for equivalent models with flow directed toward the lower plate. This systematic study provides a better understanding of the effect of mantle drag on plate displacements and deformation along subduction zones, leading to a better understanding of the ingredients required to form Andean-type mountain ranges.