  Let $K$ be a non-archimedean local field, $X$ a smooth and proper $K$-scheme,
and fix a pluricanonical form on $X$. For every finite extension $K'$ of $K$,
the pluricanonical form induces a measure on the $K'$-analytic manifold
$X(K')$. We prove that, when $K'$ runs through all finite tame extensions of
$K$, suitable normalizations of the pushforwards of these measures to the
Berkovich analytification of $X$ converge to a Lebesgue-type measure on the
temperate part of the Kontsevich--Soibelman skeleton, assuming the existence of
a strict normal crossings model for $X$. We also prove a similar result for all
finite extensions $K'$ under the assumption that $X$ has a log smooth model.
This is a non-archimedean counterpart of analogous results for volume forms on
degenerating complex Calabi--Yau manifolds by Boucksom and the first-named
author. Along the way, we develop a general theory of Lebesgue measures on
Berkovich skeleta over discretely valued fields.
