  Let $G$ be a closed highly homogeneous subgroup of $S_{\infty}$ not involving
circular orderings. We show that the closure of a conjugacy class from $G$
contains a conjugacy class which is comeagre in it. Furthermore, we show that
the family of finite partial maps extendable to elements of this conjugacy
class has the cofinal amalgamation property. Similar statements are proved for
automorphisms of typical ultrahomogeneous partial orderings.
