We employ model protocell networks for evaluation of molecular transport through lipid nanotubes as potential means of communication among primitive cells on the early Earth. Network formation is initiated by deposition of multilamellar lipid reservoirs onto a silicon oxide surface in an aqueous environment. These reservoirs autonomously develop into surface-adhered protocells interconnected via lipid nanotubes, and encapsulate solutes from the ambient buffer in the process. We prepare networks in the presence of DNA and RNA and observe encapsulation of these molecules, and their diffusive transport between the lipid compartments via the interconnecting nanotubes. By means of an analytical model we determine key physical parameters affecting the transport, such as nanotube diameter and compartment size. We conclude that nanotube-mediated transport in self-organized nanotube-vesicle networks could have been a possible pathway of chemical communication between primitive, self-assembled protocells under early earth conditions, circumventing the necessity for crossing the membrane barrier. We suggest this transport within a closed protocell network as a feasible means of RNA and DNA exchange under primitive prebiotic conditions, possibly facilitating early replication.