A Cn-move is a local move
on links defined by Habiro and Goussarov, which can be regarded as a ‘higher
order crossing change’. We use Milnor invariants with repeating indices to
provide several classification results for links up to Cn-moves, under certain
restrictions. Namely, we give a classification up to C4-moves of 2-component
links, 3-component Brunnian links and n-component C3-trivial links. We
also classify n-component link-homotopically trivial Brunnian links up to
Cn+1-moves.