The transport properties of quantum dot (QD) systems based on double-walled carbon nanotubes (DWCNTs) are investigated. The interplay between microscopic structure and strong Coulomb interaction is treated within a bosonization framework. The linear and nonlinear current-voltage characteristics of the QD system are calculated by starting from the Liouville equation for the reduced density matrix. Depending on the intershell couplings, an eight-electron periodicity of the Coulomb blockade peak spacing in the case of commensurate DWCNT QDs and a four-electron periodicity in the incommensurate case are predicted. The contribution of excited states of DWCNTs to the nonlinear transport is investigated as well.