Constriction in the flow passage of the physiological circulatory system is central to the occurrence of several diseased conditions such as thrombosis and is also pivotal towards the understanding of several regulatory processes in the human microvasculature. It is, therefore, imperative to advance a mechanistic insight into the dynamics of lipid vesicles, cell mimicking, fluid-filled compartments, through a physiologically relevant microconfinement, with particular focus on deciphering the role of its mechanophysical properties. Here we bring out the role of membrane bending rigidity and the initial deflation (deviation in shape from sphericity) on the transient shape evolution of a lipid vesicle as it migrates through a microfluidic constriction, a paradigm that is unexplored thus far. Based on our experimental observations as well as theoretical insights, we construct a regime map to elucidate the range of the key dimensionless parameters orchestrating this dynamical transition. Furthermore, our observations on the vesicle's stretching dynamics emerging from selective mapping with viscosity contrast between the encapsulated and the suspending fluid medium offer potential physiologically relevant cues on the impact of cell aging on its deformability across a constricted path. Such mechanistic insights may help in establishing quantitative correlations between the dynamical transition of a lipid vesicle and its membrane mechanics, which may in turn have decisive implications in health and disease while circulating across microvascular fluidic pathways.