Water/oil/water (w/o/w) double emulsions (DEs) are multicompartment structures which can be used in many technological applications and in fundamental studies as models of cell like microreactors or templates for other materials. Herein, we study the flow dynamics of water/oil/water double emulsions generated in a microfluidic device and stabilized with the phospholipid 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC). We show that by varying the concentration of lipids in the oil phase (chloroform) or by modulating the viscosity of the aqueous continuous phase, the double emulsions under flow exhibit a rich dynamic behavior. An initial deformation of the double emulsions is followed by tube extraction at the rear end, relative to the flow direction, resulting in pinch off at the tube extremity by which small aqueous compartments are released. These compartments are phospholipid vesicles as deduced from fluorescence experiments. The overall process can thus be of help to shed light on the mechanical aspects of phenomena such as the budding and fusion in cell membranes.