  Let $k$ be a field of characteristic $p>0$. Let $D_m$ be a $\BT_m$ over $k$
(i.e., an $m$-truncated Barsotti--Tate group over $k$). Let $S$ be a\break
$k$-scheme and let $X$ be a $\BT_m$ over $S$. Let $S_{D_m}(X)$ be the subscheme
of $S$ which describes the locus where $X$ is locally for the fppf topology
isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}(X)$ is pure in $S$
i.e., the immersion $S_{D_m}(X) \hookrightarrow S$ is affine. For
$p\in\{2,3\}$, we prove purity if $D_m$ satisfies a certain property depending
only on its $p$-torsion $D_m[p]$. For $p\ge 5$, we apply the developed
techniques to show that all level $m$ stratifications associated to Shimura
varieties of Hodge type are pure.
