  We develop a general diagrammatic theory of welded graphs, and provide an
extension of Satoh's Tube map from welded graphs to ribbon surface-links. As a
topological application, we obtain a complete link-homotopy classification of
so-called \emph{knotted punctured spheres} in $4$--space, by means of the
$4$--dimensional Milnor invariants introduced by the authors in \cite{cutAMY}.
On the algebraic side, we show that the theory of welded graphs can be
reinterpreted as a theory of Wirtinger group presentations, up to a natural set
of transformations; these groups arise as the fundamental group of the exterior
of the surface-link obtained from the given welded graph by the extended Tube
map. Finally, we address the injectivity question for the Tube map, identifying
a new family of local moves on welded links, called $\Upsilon$ moves, under
which the (non extended) Tube map is invariant.
