A palaeogeographical reconstruction of the South American and African continents back to anomaly C34 (84 Ma) brings together the Rio Grande Rise (RGR) and the central portion of the Walvis Ridge (WR), thus the RGR–WR aseismic ridges may have a common origin. If the construction of the RGR–WR basaltic plateau took place mainly between 89 and 78 Ma, as indicated by the ages of the basalts sampled by DSDP wells, then the basaltic magmas are the result of an ‘on-ridge’ volcanism. Once separated, the normal sea-floor spreading and thermal subsidence of the RGR and WR ridges continued until approximately 47 Ma when an Eocene magmatism took place in the RGR. In the WR, a younger volcanism is observed in the Guyot Province. The available geochemical and isotope data of the WR–RGR basalts do not indicate the participation of the continental crust melting component. Incompatible trace element ratios and isotope signatures of the basalts from the RGR–WR ridges are distinct from the present-day Tristan da Cunha alkaline rocks, and are nearly identical to the high-Ti Paraná Magmatic Province (PMP) tholeiites (133–132 Ma). Both the high-Ti PMP and the WR–RGR basalts are characterized by moderate initial ⁸⁷Sr/⁸⁶Sr and low ²⁰⁶Pb/²⁰⁴Pb isotope ratios [Enriched Mantle I (EMI) mantle component], suggesting melting from a common source, with significant participation of sub-continental lithospheric mantle (SCLM). A three-dimensional (3D) flexural modelling of the RGR and WR was conducted using ETOPO1 digital topography/bathymetry and EGM2008-derived free-air anomalies as a constraint. The best fit between the observed and calculated free-air anomalies was obtained for an elastic plate with elastic plate thickness (T_e) of less than 5 km, consistent with an ‘on-ridge’ initial construction of the RGR–WR. The modelling of the crust–mantle interface depths indicates a total crustal thickness of up to 30 km in the RGR–WR. Flexural analysis reinforces the geological evidence that RGR was constructed during two main magmatic episodes, the tholeiitic basalts in the Santonian–Conician times and the alkaline magmatism in the Eocene. Geochemical and geophysical evidence, which rules out the classical deep-mantle plume model in explaining the generation of basalts of these volcanic provinces, is presented. Finally, three models to explain the geochemical and isotope signatures of RGR–WR basalts are reviewed: (1) thermal erosion of SCLM owing to edge-driven convection; (2) melting of fragmented or detached SCLM and lower crust; and (3) thermal erosion at the base of the SCLM with lateral transport of enriched components by mantle flow.