The Maritime Continent experiences some of the world’s most severe convective rainfall, with an intense diurnal cycle. Akey feature is offshore propagation of convection overnight, having peaked over land during the evening. Existing hypotheses suggest this propagation is due to the nocturnal land breeze and environmental wind causing low-level convergence; and/or gravity waves triggering convection as they propagate. We use a convection-permitting configuration of the Met Office Unified Model over Sumatra to test these hypotheses, verifying against observations from the Japanese Years of the Maritime Continent field campaign. In selected case studies there is an organized squall line propagating with the land breeze density current, possibly reinforced by convective cold pools, at ∼3 m s−1 to around 150–300 km offshore. Propagation at these speeds is also seen in a composite mean diurnal cycle. The density current is verified by observations, with offshore low-level wind and virtual potential temperature showing a rapid decrease consistent with a density current front, accompanied by rainfall. Gravity waves are identified in the model with a typical phase speed of 16 m s−1. They trigger isolated cells of convection, usually further offshore and with much weaker precipitation than the squall line. Occasionally, the isolated convection may deepen and the rainfall intensify, if the gravity wave interacts with a substantial pre-existing perturbation such as shallow cloud. The localized convection triggered by gravity waves does not generally propagate at the wave’s own speed, but this phenomenon may appear as propagation along a wave trajectory in a composite that averages over many days of the diurnal cycle.