To spread from a localized tumor, metastatic cancer cells must squeeze through constrictions that cause major nuclear deformations. Since chromosome structure affects nucleus stiffness, gene regulation, and DNA repair, here, we investigate the relationship between 3D genome structure and constricted migration in cancer cells. Using melanoma (A375) cells, we identify phenotypic differences in cells that have undergone multiple rounds of constricted migration. These cells display a stably higher migration efficiency, elongated morphology, and differences in the distribution of Lamin A/C and heterochromatin. Hi‐C experiments reveal differences in chromosome spatial compartmentalization specific to cells that have passed through constrictions and related alterations in expression of genes associated with migration and metastasis. Certain features of the 3D genome structure changes, such as a loss of B compartment interaction strength, are consistently observed after constricted migration in clonal populations of A375 cells and in MDA‐MB‐231 breast cancer cells. Our observations suggest that consistent types of chromosome structure changes are induced or selected by passage through constrictions and that these may epigenetically encode stable differences in gene expression and cellular migration phenotype.