Understanding how soft matter deforms in response to mechanical interactions is central to the design of functional synthetic materials as well as elucidation of the behaviors of biological assemblies. Here we explore how cycles of thermally induced transitions between nematic (N) and isotropic (I) phases can be used to exert cyclical elastic stresses on dispersions of giant unilamellar vesicles (GUVs) and thereby evolve GUV shape and properties. The measurements were enabled by the finding that I–N–I phase transitions of the lyotropic chromonic liquid crystal disodium cromoglycate, when conducted via an intermediate columnar (M) phase, minimized transport of GUVs on phase fronts to confining surfaces. Whereas I to N phase transitions strained spherical GUVs into spindle-like shapes, with an efflux of GUV internal volume, subsequent N to I transitions generated a range of complex GUV shapes, including stomatocyte, pear- and dumbbell-like shapes that depended on the extent of strain in the N phase. The highest strained GUVs were observed to form buds (daughter vesicles) that we show, via a cycle of I–N–I–N phase transitions, are connected via a neck to the parent vesicle. Additional experiments established that changes in elasticity of the phase surrounding the GUVs and not thermal expansion of membranes were responsible for the shape transitions, and that I–N–I transformations that generate stomatocytes can be understood from the Bilayer-Coupling model of GUV shapes. Overall, these observations advance our understanding of how LC elastic stresses can be regulated to evolve the shapes of soft biological assemblies as well as provide new approaches for engineering synthetic soft matter.