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 lipid reservoirs onto a SiO2 surface in an aqueous environment. These reservoirs autonomously develop into surface-adhered protocells interconnected via lipid nanotubes while encapsulating solutes from the ambient buffer. We observe the uptake of DNA and RNA, 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 could have been a possible pathway of communication between primitive cells on the early Earth, circumventing the necessity for crossing the membrane barrier. We suggest this transport as a feasible means of RNA and DNA exchange under primitive prebiotic conditions, possibly facilitating early replication.