Several authors have proposed that granitic melt accumulation and transport from the source region occurs in networks of connected melt-filled veins and dykes. These models envisage the smallest leucosomes as ‘rivulets’ that connect to feed larger dykes that form the ‘rivers’ through which magma ascends through the sub-solidus crust. This paper critically reviews this ‘rivulets-feeding-rivers’ model. It is argued that such melt-filled networks are unlikely to develop in nature, because melt flows and accumulates well before a fully connected network can be established. In the alternative stepwise accumulation model, flow and accumulation is transient in both space and time. Observations on migmatites at Port Navalo, France, that were used to support the existence of melt-filled networks are discussed and reinterpreted. In this interpretation, the structures in these migmatites are consistent with the collapse and draining of individual melt batches, supporting the stepwise accumulation model.