Cells that migrate through small, rigid pores and that have normal levels of the nuclear structure protein lamin-A exhibit an increase in DNA damage, which is also observed with lamin-A depletion in diseases such as cancer and with many lamin-A mutations. Here we show nuclear envelope rupture is a shared feature that increases in standard culture after lamin-A knockdown, which causes nuclear loss of multiple DNA repair factors and increased DNA damage. Some repair factors are merely mis-localized to cytoplasm whereas others are partially depleted unless rescued by lamin-A expression. Compared to standard cultures on rigid glass coverslips, the growth of lamin-A low cells on soft matrix relaxes cytoskeletal stress on the nucleus, suppresses the mis-localization of DNA repair factors, and minimizes DNA damage nearly to wildtype levels. Conversely, constricted migration of the lamin-A low cells causes abnormally high levels of DNA damage, consistent with sustained loss of repair factors. The findings add insight into why monogenic progeroid syndromes that often associate with increased DNA damage and predominantly impact cells in stiff tissues result from mutations only in lamin-A or DNA repair factors.