The numerical study of 3D vesicles with a reduced volume equal to that of human red blood cells leads to the discovery of three types of dynamics: (i) squaring motion, in which the angle between the direction of the longest distance and the flow velocity undergoes discontinuous jumps over time, (ii) spontaneous parity breaking of the shape leading to cross-streamline migration, and (iii) S tumbling where the vesicle tumbles, exhibiting a pronounced S-like shape with a waisted morphology in the center. We report on the phase diagram within a wide range of relevant parameters. Our estimates reveal that healthy and pathological red blood cells are also prone to these types of motion, which may affect blood microcirculation and impact oxygen transport.