Cytoskeletal filaments and motor proteins are critical components in the transport and reorganization of membrane-based organelles in eukaryotic cells. Previous studies have recapitulated the microtubule-kinesin transport system in vitro to dynamically assemble large-scale nanotube networks from multilamellar liposomes and polymersomes. Moving toward more biologically relevant systems, the present work examines whether lipid nanotube (LNT) networks can be generated from giant unilamellar vesicles (GUVs) and subsequently characterizes how the lipid composition may be tuned to alter the dynamics, structure, and fluidity of networks. Here, we describe a two-step process in which microtubule motility (i) drives the transport and aggregation of GUVs to form structures with a decreased energy barrier for LNT formation and (ii) extrudes LNTs without destroying parent GUVs, allowing for the formation of large LNT networks. We further show that the lipid composition of the GUV influences formation and morphology of the extruded LNTs and associated networks. For example, LNTs formed from phase-separated GUVs (e.g., liquid-solid phase-separated and coexisting liquid-ordered and liquid-disordered phase-separated) display morphologies related to the specific phase behavior reflective of the parent GUVs. Overall, the ability to form nanotubes from compositionally complex vesicles opens the door to generating lipid networks that more closely mimic the structure and function of those found in cellular systems.