  In this paper we introduce the definition of $(k,l)$-universal transversal
property, which is a refinement of the definition of $k$-universal transversal
property, which in turn is a refinement of the classic definition of
$k$-homogeneity for permutation groups. In particular, a group possesses the
$(2,n)$-universal transversal property if and only if it is primitive; it
possesses the $(2,2)$-universal transversal property if and only if it is
$2$-homogeneous. Up to a few undecided cases, we give a classification of
groups satisfying the $(k,l)$-universal transversal property, for $k\ge 3$.
Then we apply this result for studying regular semigroups of partial
transformations.
