The Southern Ocean plays an integral role in the global climate system, exchanging heat, salt, and carbon throughout the major ocean basins via the deep, fast-flowing Antarctic Circumpolar Current. The Antarctic Circumpolar Current is bounded by spatially and temporally varying fronts that partition distinct water masses. Locating and quantifying cross-front transport is crucial for understanding global patterns of inter-basin exchange; however, this is challenging because fronts are typically defined by hydrographic properties, such as temperature or salinity, which are subject to external sources and sinks, rather than by properties of the flow itself. Here we show that, when characterized by material contours that minimize deformation and cross-contour mixing, Southern Ocean fronts exhibit a global pattern of alternating poleward and equatorward transport caused by frontal meandering, which, in turn is influenced by prominent sea-floor obstacles. These results highlight the importance of bathymetric features in controlling Southern Ocean dynamics and inter-basin exchange.