The cosmological applications of atomic clocks1, 2, 3 so far have been limited to searches for the uniform-in-time drift of fundamental constants4. We point out that a transient-in-time change of fundamental constants can be induced by dark-matter objects that have large spatial extent, such as stable topological defects5 built from light non-Standard Model fields. Networks of correlated atomic clocks, some of them already in existence6, such as the Global Positioning System, can be used as a powerful tool to search for topological defect dark matter, thus providing another important fundamental physics application for the ever-improving accuracy of atomic clocks. During the encounter with an extended dark-matter object, as it sweeps through the network, initially synchronized clocks will become desynchronized. Time discrepancies between spatially separated clocks are expected to exhibit a distinct signature, encoding the defect’s space structure and its interaction strength with atoms.