  We use the elliptic interpolation kernel due to the second author to prove an
$\mathrm{A}_n$ extension of the elliptic Selberg integral. More generally, we
obtain elliptic analogues of the $\mathrm{A}_n$ Kadell, Hua-Kadell and
Alba-Fateev-Litvinov-Tarnopolsky (or AFLT) integrals.
