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Level $m$-stratifications on PEL Shimura varieties are defined and studied byWedhorn using BT-$m$s with PEL structure, and then by Vasiu for general Hodgetype Shimura varieties using Shimura $F$-crystals. The theory of foliations isestablished by Oort for Siegel modular varieties, and by Mantovan for PELShimura varieties. It plays an important role in Hamacher's work to compute thedimension of Newton strata of PEL Shimura varieties.  We study level $m$ stratifications on good reductions at $p>2$ of Shimuravarieties of Hodge type by constructing certain torsors together withequivariant morphisms, and relating them to truncated displays. We then use theresults obtained to extend the theory of foliations to these reductions. As aconsequence, combined with results of Nie and Zhu, we get a dimension formulafor Newton strata.

Stratifications and foliations for good reductions of Shimura varieties of Hodge type