Varieties swept out by grassmannians of lines

We classify complex projective varieties of dimension $2r geq 8$ swept out by a family of codimension two grassmannians of lines $mathbb{G}(1,r)$. They are either fibrations onto normal surfaces such that the general fibers are isomorphic to $G(1,r)$ or the grassmannian $mathbb{G}(1,r+1)$. The cases $r=2$ and $r=3$ are also considered in the more general context of varieties swept out by codimension two linear spaces or quadrics.