  In the paper we introduce new metric structures on $\mathfrak{g}$-foliations
that are less rigid than the well-known structures: almost contact and
3-quasi-Sasakian structures as well as $f$-structures with parallelizable
kernel and almost para-$\phi$-structures with complemented frames. We discuss
the properties of the new structures in order to demonstrate similarities with
the corresponding classical structures. Then using the flow of metrics on a
$\mathfrak{g}$-foliation, we build deformation retraction of our structures
with positive partial Ricci curvature onto the subspace of the aforementioned
classical structures.
