  Every linear set in a Galois space is the projection of a subgeometry, and
most known characterizations of linear sets are given under this point of view.
For instance, scattered linear sets of pseudoregulus type are obtained by
considering a Desarguesian spread of a subgeometry and projecting from a vertex
which is spanned by all but two director spaces. In this paper we introduce the
concept of linear sets of $h$-pseudoregulus type, which turns out to be
projected from the span of an arbitrary number of director spaces of a
Desarguesian spread of a subgeometry. Among these linear sets, we characterize
those which are $h$-scattered and solve the equivalence problem between them; a
key role is played by an algebraic tool recently introduced in the literature
and known as Moore exponent set. As a byproduct, we classify asymptotically
$h$-scattered linear sets of $h$-pseudoregulus type.
