The influence of tectonics and climate on basin-fill erosion and incision in the Gulf of Corinth rift, central Greece, and Rio Grande rift, southwest USA, is examined. Overall, it is suggested that climate change controls the downstream water:sediment ratio and sediment transport capacity, via operation of the continuity equation. Tectonics, specifically the rapid growth and propagation of structures, sets up gradient contrasts and upstream-migrating changes in transport capacity via operation of the diffusivity equation. A steady-state aggradational mode operated in the southern Rio Grande rift between ~5 and 0.8 Ma, causing preservation of ancestral axial-channel and floodplain deposits due to relatively slow, long-term active rift subsidence. The onset of major climatic change around 0.8 Ma resulted in the axial river periodically incising to a total extent of ~150 m, removing about 25% by volume of previously accumulated sediment, despite continued active faulting and fault-induced subsidence. This climatic mode is interpreted to be a periodic response to positive downstream gradients in sediment transport rate during glacial and glacial-transition periods, caused by low-level external sediment sourcing and a dominance of large magnitude spring snowmelt floods from northern mountain valleys. Tectonic drivers are spectacularly demonstrated in the Gulf of Corinth, where a new theory of 'piggy-back' basin abandonment and regional uplift is proposed, as formerly active rift-margin faults are progressively dragged above the flat slab of underriding African lithosphere. Basin abandonment occurred across newly propagating faults, with erosion and basin-fill incision of up to 800 m depth, as discontinuities in drainage channel slope have migrated rapidly upstream. In both rifts, sediment relaxation time, T s = l 2 /κ, where l is a length scale and κ is sediment diffusivity, is probably short, since relevant length scales are small and diffusivities large. Thus in the Rio Grande rift, despite the great length of the river system as a whole, it is the balance between hydrological and sediment input from the many lateral tributaries that controls non-uniform transport capacity of the axial channel. In the case of the Gulf of Corinth rift, it is the high strain rate that causes diffusivity to be large in drainages cutting across rapidly vertically-growing normal faults.