Luna's fundamental lemma for diagonalizable groups

We study relatively affine actions of a diagonalizable group $G$ on locally noetherian schemes. In particular, we generalize Luna's fundamental lemma when applied to a diagonalizable group: we obtain criteria for a $G$-equivariant morphism $f: X'\to X$ to be $strongly\ equivariant$, namely the base change of the morphism $f/\!/G$ of quotient schemes, and establish descent criteria for $f/\!/G$ to be an open embedding, \'etale, smooth, regular, syntomic, or lci.