  We consider knotted annuli in 4-space, called 2-string-links, which are
knotted surfaces in codimension two that are naturally related, via closure
operations, to both 2-links and 2-torus links. We classify 2-string-links up to
link-homotopy by means of a 4-dimensional version of Milnor invariants. The key
to our proof is that any 2-string link is link-homotopic to a ribbon one; this
allows to use the homotopy classification obtained in the ribbon case by P.
Bellingeri and the authors. Along the way, we give a Roseman-type result for
immersed surfaces in 4-space. We also discuss the case of ribbon k-string
links, for $k\geq 3$.
