It will work, but going from 6K to 10 or 12K is a very huge step, and nothing I would recommend.

Basically after the bevel is set the aim is not to remove metal fast, but to refine the edge to the sharpest and smoothest possible by graually remove the scratches from the previous stone.
A 6K stone leaves much deeper scratches than a 8K, so the amount of additional effort to remove those scratches on a 12K would be significant.