The interaction of mantle plume driven flow with upwelling flow due to a nearby mid‐ocean ridge occurs for many mantle plumes including Galápagos and Iceland. This interaction is typified by trace element and isotopic signatures demonstrating the “contamination” of normal ridge composition by relatively enriched plume material. However, another common signature of plume‐ridge interaction is volcanic lineaments linking ridges and nearby plumes, perhaps most conspicuously the Wolf‐Darwin lineament (WDL) at Galápagos and the Rodrigues Ridge (RR) at La Réunion. These enigmatic features remain unexplained. Plume‐ridge interaction is commonly modeled in terms of interaction between solid‐state plume flow and divergent ridge flow, but such models do not likely lead to the kind of solid‐state flow channelization that might explain narrow features such as the WDL and RR. Likewise, models involving tapping of anomalously hot and/or fertile asthenosphere between the plume and ridge due to lithospheric faulting appear to be inconsistent with a variety of evidence. We propose an alternative model in which the lineaments are the surface expressions of localized melt channels in the asthenosphere formed due to instabilities in a two‐phase partially molten system. A thermodynamic analysis shows that given the magma fluxes inferred to be associated with structures such as WDL and RR, these melt channels can be maintained over plume‐ridge distances up to ∼1000 km. These results suggest that plume‐ridge interaction in general, possibly including transport of plume‐derived material along ridge axes (e.g., Iceland), may involve transport in high‐melt‐fraction channels, as opposed to just solid‐state mantle flow.