  We study homotopy groups of spaces of links, focusing on long links of
codimension at least three. In the case of multiple components, they admit
split injections from homotopy groups of spheres. We show that, up to knotting,
these account for all the homotopy groups in a range which depends on the
dimensions of the source manifolds and target manifold and which roughly
generalizes the triple-point-free range for isotopy classes. Just beyond this
range, joining components sends both a parametrized long Borromean rings class
and a Hopf fibration to a generator of the first nontrivial homotopy group of
the space of long knots. For spaces of equidimensional long links of most
source dimensions, we describe generators for the homotopy group in this degree
in terms of these Borromean rings and homotopy groups of spheres. A key
ingredient in most of our results is a graphing map which increases source and
target dimensions by one.
