<p>The South American subduction zone remains a topic of debate with long-lasting questions involving the origin of non-collisional orogeny and the effect of very large trench-parallel extent, slab sinking to great mantle depths, and aseismic ridge subduction. A key to help solve those issues is through studying the subduction zone dynamics with buoyancy-driven numerical modelling that uses constrained independent variables in order to best approximate the dynamics of the real subduction system. We conduct a parametric investigation on the effect of upper mantle rheology (Newtonian or non-Newtonian), subduction interface yield stress and slab thermal weakening. As a means of constraining those model variables we attempt to find best-fits by comparing our model outcomes with the present-day upper- and lower-mantle slab geometry observed on tomography models and obtained from earthquake hypocentre locations, as well as with estimates of Cenozoic velocities obtained from kinematic reconstruction. Key ingredients that need to be reproduced are slab flattening close to the surface, strong oscillation of the Farallon-Nazca subducting plate velocity and progressive decrease in trench retreat rate after a long period of time. We include these ingredients to define a model fitting score that contains a total of 9 criteria. Our best fitting model involves significant slab thermal weakening in order to attain the fast Farallon-Nazca subducting plate velocity and to better reproduce the subduction partitioning in the past 48 Myr, due to strong reduction of the shear stresses resisting downdip slab sinking and of the slab bending resistance. We further find that a non-Newtonian upper mantle rheology promotes slab folding and realistic associated oscillation of the subducting plate velocity. Our parametric study also indicates that the subduction interface must be weak in agreement with earlier laboratory subduction models, but not too weak, with a yield stress of ~ 14–21 MPa, otherwise the fit becomes poor. Our models moreover suggest that slab folding at the 660 km discontinuity can be a cause of the Farallon-Nazca subducting plate velocity oscillation. Whether and how this slab folding process induces periodic/episodic variations in deformation of the Andes remains an open question that requires further research.</p>