Blueing is essentially controlled “rust”’
Back in the “old days”, the process involved getting the required surfaces RUSTY, i.e. BROWN, for a start. This is why many Brit technical documents refer to the process as “browning”, not “blueing”
Being RUST, the quickest way to remove it is with a “rust-remover”, as available from your local hardware shop.
The original process was very labour-intensive. The idea was to get all of the appropriate suraces covered with an even film of basic, brown rust, and then “card” it off with soft steel brushes and, literally, “card”.
Once the rust was “carded", it went back into the steam room and allowed to go rusty again. After each such cycle, the colour becomes darker, thicker and more “lustrous”.
About the only place you will find folk doing this these days is in “bespoke” (custom) gunmakers and very “up-market” gunsmith shops. It is VERY labour-intensive., and thus EXPENSIVE
These days, you just polish everything to the required “sheen” and run it through the “Dulite” tank.
HOWEVER, components made from exotic alloy steels form very interesting surface films when heat-treated. If such components are just whacked in the blueing tank with everything else, they MAY come out being anything from grey to purple.
There are two tricks:
1. Ultra-fine bead blast the offending parts before blueing; they will come out (if everything else is OK), a lustrous, satin black.
2. If you want the “full-gloss” look, drop the temperature of the tank ten degrees and immerse the part(s). They should start to develop the desired dark colour. Then slowly raise the temperature to “normal” to finish the job. It can take a bit of experimenting.
"Parkerising" is vaguely related, but the biggest difference is that the metal surface MUST be "activated", i.e. grit blasted. If this is not done before immersion in the phosphating tank, the usual result is a blotchy, pallid, grey look, the rifle looks "hung-over".
The phosphating MUST be done as soon as possible after grit-blasting; if left too long, the bare metal will start to oxidise and the end result will be uneven. After BOTH blueing and phosphating, the metalwork MUST be "de-watered" and thoroughly oiled.
Ultimately, BOTH processes are done to provide a hard, fine, crystalline film that HOLDS OIL in order to prevent corrosion; that it looks "nice" is a secondary consideration.