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Your question is strange in that it considers both cases where the characteristic polynomial changes and where it does not; see my comments to Richards Stanley's answer. It seems like if you break up a multiple root of the characteristic polynomial into smaller groups, each corresponding to a new root, then you can distribute each Jordan block size for the original eigenvalue into at most one block for each new eigenvalue. Then for any partitions into blocks so obtained for an eigenvalue (old or new), also take everything above it in the dominance order, to reflect the closure relation among nilpotent orbits. Just how many cases this gives is not so clear to me; you do not seem to care which newly created eigenvalue is close to which old one in distinguishing patterns, and this might affect the outcome.