This paper is a continuation of a paper by de Shalit and Goren from 2018. We study foliations of two types on Shimura varieties
S
S
in characteristic
p
p
. The first, which we call tautological foliations, are defined on Hilbert modular varieties, and lift to characteristic
0
0
. The second, the
V
V
-foliations, are defined on unitary Shimura varieties in characteristic
p
p
only, and generalize the foliations studied by us before, when the CM field in question was quadratic imaginary. We determine when these foliations are
p
p
-closed, and the locus where they are smooth. Where not smooth, we construct a successive blowup of our Shimura variety to which they extend as smooth foliations. We discuss some integral varieties of the foliations. We relate the quotient of
S
S
by the foliation to a purely inseparable map from a certain component of another Shimura variety of the same type, with parahoric level structure at
p
p
, to
S
.
S.