During Drosophila cellularization, 6000 nuclei migrate to the plasma membrane to form an epithelial sheet. To simultaneously invaginate the new cell membranes, the embryonic membrane must expand 25x from a reservoir of microvilli. Exocytosed, Golgi-derived vesicles resupply the reservoir during the first 30 minutes of membrane invagination. Concomitantly with the completion of exocytosis, velocity jumps ~6x, and elastic and viscous properties drop. The governing mechanisms are experimentally occluded by the redundancy of motor proteins, microtubules and F-actin. A mathematical model can disentangle the complexity of this process and shed light on possible mechanisms. In this work, we propose a model of furrow invagination by representing the membrane-cytoskeleton composite with a viscoelastic model, constrained by conservation of area, and dynamic cytoskeletal properties. We observe that our continuum model is capable of governing the slow-to-fast transition and the final furrow length. We expect the model will lead to new insights into the biophysical invagination mechanisms. While our current focus is on Drosophila cellularization, this model may be extended to understand other multinucleated systems, such as T-tubule formation in muscle cells.