You can damascus finish to Brown & White or Black & White.
The difference in the finishing process will be that in the Brown& White there will be no boiling water step. You will be leaving the color as is...brown rust.

In the Black & White finish,,you will be boiling the rusted bbl's to convert the red rust to black rust (as in Rust bluing).

To get the contrast of the White 'color',,you will need to etch the bbls betw the rusting cycles.
This loosens the rust you just allowed to form. BUT, the key here is that the etching process loosens the rust on the STEEL component of the Damascus bbl quicker than on the IRON component of the safe barrel.
Damascus being a composite of both Iron and Steel forge welded together from strips of the two metals.

The etching soln most commonly used is Ferric Chloride.
That is a common chemical used to etch copper circuit boards.
It can be bought inexpensively off the net from suppliers. ebay is one place where severl suppliers sell it.

The ething soln is made with plain water,,tap water is fine.
The soln is a very weak one. I use one of about 5% Ferric Chloride to Water.
Mixing it doesn't need to be Lab precise. If it's weaker, that's fine too. I would not go much if any stronger than 5% Ferric Chloride to water.
Use it at room temp. NEVER warm the etch in using it. It works VERY fast as it is.

To use all this stuff..
***Start by first protecting the bores which is needed because of the use of the etchng soln.
This is explained at the bottom *****

Then..
Start by doing a standard Slow Rust Brown on the set of bbls.
I use Lauer Mtn Slow Rust Soln for both Rust Brown and Rust Bluing finish. I've used it for about 40yrs. Others have their favorite stuff.
Use what you are comfortable with. You are Slow Rust Browning/ Slow Rust Bluing,,the only added step is the etching to bring out the damascus pattern.

Once they have gotten a suitible coating of rust on them, now you decide if you want a Brown/White final color,,or a Black/White final color.

If Brown/White,,you are going to card the bbls now.

If you are wanting a Black&White final look,,you will now boil the bbls just as is you are rust bluing them. Then card them as usual for a rust blue.
Then let the bbls cool to room temp.

Next is the etching process.
Your etch soln is easiest stored in a PVC tube with a cap PVC glued on one end, and a PVC cap simply set on the other.
I tie the tube to one end of my bench and it stays there until I need to use it. Then I bring it over to the washtubs and re-tie it into position for use.
The tube must be long enough to allow the longest set of bbls to be completley submerged.
Remember when filling the tube with the etching soln to leave some room for the soln to rise in level when the bbls go in so you don't have over flow.

Take the room temp set of bbl's and hook a wire around the breech end (easiest to find something to hand on to.) Coat Hanger wire works good. Strong and won't suddenly leave you with a piece of wire in your hand and a set of bbls deep in the tube soaking in Ferric Chloride.

Dunk the bbl's into and back out of the etch in one complete motion. Down in and right back out.
No waiting,,no hesitation.
Once the bbls are down in and submerged,,pull them right back out.
Then right over to the washtubs (if you have them) with no hesitation and with the cool water running,,flush off the bbls of the etching soln and immedietly begin carding the surfaces with 0000 steel wool WITH the water running over them as you card the surfaces.

This will remove the color from the Steel portion of the bbl surface and leave Most of the color built up on the softer Iron surfaces.

It's a very slow process , kind of a 3 steps forward and 2 steps back each time you complete a cycle as you do remove a bit of the color from the softer iron as well,,can't be helped.
The first few cycles you won't see much of any results generally. But you are building the color in the bbl mtr'l. It will start to show as you repeat cycles.

Some finishers build the color by doing multiple cycles of brown or black and only then do a single Etching cycle. Maybe 3 or 4 and then 1 etching,,then 3 or 4 more color cycles and then one etching,,etc
They feel it builds the damascus finish quicker. Others etch with each color cycle (I do it that way)
Try it both ways and see what results you get.

Regular rust bluing can be done in as few as 5 or 6 cycles and have a nice finish.
Damascus finishing with it's etching with removes some of the finish each time requires many more cycles to completion.
I find it usually takes 15 to 20 cycles to get a nice final damascus finish. But sometimes you are rewarded with a nice finish in less cycles.

Bbl composition makes a difference I'm sure. Some rusting formulas work better on some bbl's than others. The weaker etching solns will leave more color on the Iron than a stonger one,,more color build up each cycle.

When you are satisfied with the final look, you can do a final very light carding with cloth,,canvas works well to burnish the look.

I like to use plain Boiled Linseed oil as the over coat oil on the finished bbls. Wiped down and then wiped off with a soft cloth or even a clean paper towel so as not to leave any excess amt on the surface.
Let them hang for a day or so afterwards.

Some use a color toned shellac or laquer as a final finish on damascus finished bbls. Very thin in consistancy and even applied it can beautifully enhance the damascus finish color and contrast. It was often used on original production damascus bbl'd guns.
This is applied to the bbls after wiping down to remove any post finishing oil that was applied of course.

****** Protecting the Bores..
....The Bores need to be protected from the etching soln during this process.
You can use wooden or rubber plugs,,many do. I don't trust them though.
I have always used plain shellac to coat the bores.
I clean them first. Then wipe the bores out with alcohol on a patch.

Then with a somewhat loose but large & long patch on a cleaning rod, I coat the bores with plain shellac. Be generous in application Do one then the other. Then back to the first one which should have dried a bit. Then back to the second one for a second coat.
Let them dry for a while. Wipe any excess that gets onto the outside metal.
Then start your rusting soln wipe on process and let the bbls hang to rust on their own.

I usually re-coat the bores with shellac right over what is there around cycle 8 or 10 just to be safe. Same process.
Ive never had a problem .
It holds up to the boiling water of rust bluing if you are doing Black&White damascus color.

Remove the shellac after you are done with bore swabs & alcohol or laq thinner, Neither will harm the new finish. You will be surprised how difficult the shellac is to remove as well.