  The main purpose of this paper is the study of a~new class of summing
multilinear operators acting from the product of Banach lattices with some
nontrivial lattice convexity. A~mixed Pietsch-Maurey-Rosenthal type
factorization theorem for these operators is proved under weaker convexity
requirements than the ones that are needed in the Maurey-Rosenthal
factorization through products of $L^q$-spaces. A~by-product of our
factorization is an extension of multilinear operators defined by
a~$q$-concavity type property to a~product of special Banach function lattices
which inherit some lattice-geometric properties of the domain spaces, as order
continuity and $p$-convexity. Factorization through Fremlin's tensor products
is also analyzed. Applications are presented to study a~special class of linear
operators between Banach function lattices that can be characterized by a
strong version of $q$-concavity. This class contains $q$-dominated operators,
and so the obtained results provide a~new factorization theorem for operators
from this class.
