First of all, note that $DSD$ and $SD^2$ have the same eigenvalues, so you are asking how the eigenvalues $\lambda_n$ of a Hermitian matrix $S$ change when it is multiplied by a diagonal matrix $D^2$ with positive diagonal elements $\delta_n^2$. As you can read in this MSE posting, the eigenvalues $\mu_n$ of $SD^2$ are related to $\lambda_n$ and $\delta_n$ by a variation on Horn's inequalities. If you order the eigenvalues as $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_K$, $\delta_1\geq\delta_2\geq\cdots\geq \delta_K$,
$\mu_1\geq\mu_2\geq\cdots\geq \mu_K$, then
$$\mu_{i+j-1}\leq \lambda_i\delta_j^2,\;\;\mu_{i+j-K}\geq\lambda_i\delta_j^2.$$