We calculate spectra of escaping cosmic rays (CRs) accelerated at shocks produced by expanding Galactic superbubbles powered by multiple supernovae producing a continuous energy outflow in star-forming galaxies. We solve the generalized Kompaneets equations adapted to expansion in various external density profiles, including exponential and power-law shapes, and take into account that escaping CRs are dominated by those around their maximum energies. We find that the escaping CR spectrum largely depends on the specific density profiles and power source properties, and the results are compared to and constrained by the observed CR spectrum. As a generic demonstration, we apply the scheme to a superbubble occurring in the centre of the Milky Way, and find that under specific parameter sets the CRs produced in our model can explain the observed CR flux and spectrum around the second knee at 10 17 eV.