In many contexts of development, regeneration, or disease such as cancer, a cell squeezes through a dense tissue or a basement membrane, constricting its nucleus. Here, we describe how the severity of nuclear deformation depends on a nucleus’ mechanical properties that are mostly determined by the density of chromatin and by the nuclear lamina. We explain how constriction-induced nuclear deformation affects nuclear contents by causing (i) local density changes in chromatin and (ii) rupture of the nuclear lamina and envelope. Both processes mislocalize diffusible nuclear factors including key DNA repair and regulatory proteins. Importantly, these effects of constricted migration are accompanied by excess DNA damage, marked by phosphorylated histone γH2AX in fixed cells. Rupture has a number of downstream consequences that include a delayed cell cycle—consistent with a damage checkpoint—and modulation of differentiation, both of which are expected to affect migration-dependent processes ranging from wound healing to tumorigenic invasion.