The nature of dark matter remains one of the greatest unsolved mysteries in elementary particle physics. It might well be that the dark matter particle belongs to a dark sector completely secluded or extremely weakly coupled to the visible sector. We demonstrate that gravitational waves arising from first-order phase transitions in the early Universe can be used to look for signatures of dark sector models connected to neutron physics. This introduces a new connection between gravitational-wave physics and nuclear physics experiments. Focusing on two particular extensions of the Standard Model with dark U(1) and SU(2) gauge groups constructed to address the neutron lifetime puzzle, we show how those signatures can be searched for in future gravitational-wave and astrometry experiments. Published by the American Physical Society 2024