  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 $\underline{\alpha}(\vec{q})$ is investigated by means of
linear response theory. A corresponding expression for
$\underline{\alpha}(\vec{q})$ 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/Fe$_{1-x}$Co$_x$/Pt)$_n$ as an
example for systems without inversion symmetry we demonstrate the occurrence of
non-vanishing linear contributions. For the alloy system bcc Fe$_{1-x}$Co$_x$
having inversion symmetry, on the other hand, only the quadratic contribution
is non-zero. 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 non-collinear spin configuration.
