  Killing of supports along subsets $\mathcal{U}$ of a group $G$ and regradings
along certain maps of groups $\phi :G'\longrightarrow G$ are studied, in the
context of group-graded algebras. We show that, under precise condition on
$\mathcal{U}$ and $\phi$, the graded module theories over the initial and the
final algebras are functorially well-connected. Special attention is paid to
$G=\mathbf{Z}$, in which case the results can be applied to $n$-Koszul
algebras.
