  Using asymptotic characterization results of spacetimes at conformal
infinity, we prove that Kerr-Schild-de Sitter spacetimes are in one-to-one
correspondence with spacetimes in the Kerr-de Sitter-like class with
conformally flat $\mathscr{I}$. Kerr-Schild-de Sitter are spacetimes of
Kerr-Schild form with de Sitter background that solve the $(\Lambda>0)$-vacuum
Einstein equations and admit a smooth conformal compactification sharing
$\mathscr{I}$ with the background metric. Kerr-de Sitter-like metrics with
conformally flat $\mathscr{I}$ are a generalization of the Kerr-de Sitter
metrics, defined originally in four spacetime dimensions and extended here to
all dimensions in terms of their initial data at null infinity. We explicitly
construct all metrics in this class as limits or analytic extensions of Kerr-de
Sitter. The structure of limits is inferred from corresponding limits of the
asymptotic data, which appear to be hard to guess from the spacetime metrics.
