I am clearly too late for the party, but I would like to mention my recent works which concern the cohomology and Chow ring of $BPGL_n$ for $n$ not necessarily a prime number:

Gu, X, On the cohomology of the classifying spaces of projective unitary groups. https://arxiv.org/abs/1612.00506

Gu, X, Some torsion classes in the Chow ring and cohomology of $BPGL_n$.
  https://arxiv.org/abs/1612.00506