We present a study on the effect of the scale of local segmental motion on an important mechanical property of glassy polymers: craze initiation via nanovoid growth. Stress-coupled positronium annihilation lifetime spectroscopy (PALS) is used for this study. A series of polyester copolymer glasses based on poly(ethylene terephthalate) (PET) and poly(1,4-cyclohexylenedimethylene terephthalate) (PCT) are used to vary the scale of local segmental motion. The results show that, in this series of copolymers, nanovoids in the copolymer containing more C-rings have more difficulty in growing. Kinetic analysis suggests that the scale of local segmental motion influences the kinetics in nanovoid growth. Larger-scale local segmental motion results in higher kinetic energy barrier and slower kinetics in nanovoid growth, which might retard craze nucleation and increase craze stress.