Is this group known outside of the stable range? If so, what is it? If not, what is known about it?



The kernel of the natural map $\pi_k(BU(r)) \to \pi_k (BU)$
For the homotopy groups of $U(r)$, hence also of $BU(r)$, you can look at the survey by M. Mimura, "Homotopy theory of Lie groups", which is chapter 19 in the "Handbook of Algebraic Topology", edited by I. M. James. It gives $\pi\_k(SU(n))$ for $k\le15$ and for $k\le 2n+6$, in section 3. The survey also lists many further references.


