Unilamellar lipid vesicles can serve as model for protocells. We present a vesicle fission mechanism in a thermal gradient under flow in a convection chamber, where vesicles cycle cold and hot regions periodically. Crucial to obtain fission of the vesicles in this scenario is a temperature-induced membrane phase transition that vesicles experience multiple times. We model the temperature gradient of the chamber with a capillary to study single vesicles on their way through the temperature gradient in an external field of shear forces. Starting in the gel-like phase the spherical vesicles are heated above their main melting temperature resulting in a dumbbell-deformation. Further downstream a temperature drop below the transition temperature induces splitting of the vesicles without further physical or chemical intervention. This mechanism also holds for less cooperative systems, as shown here for a lipid alloy with a broad transition temperature width of 8 K. We find a critical tether length that can be understood from the transition width and the locally applied temperature gradient. This combination of a temperature-induced membrane phase transition and realistic flow scenarios as given e.g. in a white smoker enable a fission mechanism that can contribute to the understanding of more advanced protocell cycles.