The modification of the magnetization dissipation or Gilbert damping caused by an inhomogeneous magnetic structure and expressed in terms of a wave vector dependent tensor α––(→q) is investigated by means of linear response theory. A corresponding expression for α––(→q) in terms of the electronic Green function has been developed giving in particular the leading contributions to the Gilbert damping linear and quadratic in q. Numerical results for realistic systems are presented that have been obtained by implementing the scheme within the framework of the fully relativistic KKR (Korringa-Kohn-Rostoker) band structure method. Using the multilayered system (Cu/Fe1−xCox/Pt)n as an example for systems without inversion symmetry we demonstrate the occurrence of nonvanishing linear contributions. For the alloy system bcc Fe1−xCox having inversion symmetry, on the other hand, only the quadratic contribution is nonzero. As it is shown, this quadratic contribution does not vanish even if the spin-orbit coupling is suppressed, i.e., it is a direct consequence of the noncollinear spin configuration.