Cancer cells' ability to migrate through constricting pores in the tissue matrix is limited by nuclear stiffness. MT1-MMP contributes to metastasis by widening matrix pores, facilitating confined migration. Here, we show that modulation of matrix pore size or of lamin A expression known to modulate nuclear stiffness directly impinges on levels of MT1-MMP-mediated pericellular collagenolysis by cancer cells. A component of this adaptive response is the centrosome-centered distribution of MT1-MMP intracellular storage compartments ahead of the nucleus. We further show that this response, including invadopodia formation in association with confining matrix fibrils, requires an intact connection between the nucleus and the centrosome via the linker of nucleoskeleton and cytoskeleton (LINC) complex protein nesprin-2 and dynein adaptor Lis1. Our results uncover a digest-on-demand strategy for nuclear translocation through constricted spaces whereby confined migration triggers polarization of MT1-MMP storage compartments and matrix proteolysis in front of the nucleus depending on nucleus-microtubule linkage.