  Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space
which bound immersed 3-manifolds with only ribbon singularities. They appear as
topological realizations of welded knotted objects, which is a natural quotient
of virtual knot theory. In this paper we consider ribbon tubes and ribbon
torus-links, which are natural analogues of string links and links,
respectively. We show how ribbon tubes naturally act on the reduced free group,
and how this action classifies ribbon tubes up to link-homotopy, that is when
allowing each component to cross itself. At the combinatorial level, this
provides a classification of welded string links up to self-virtualization.
This generalizes a result of Habegger and Lin on usual string links, and the
above-mentioned action on the reduced free group can be refined to a general
"virtual extension" of Milnor invariants. As an application, we obtain a
classification of ribbon torus-links up to link-homotopy.
