Please note the Damascus patterns are brought out by the browning/bluing process. Usually the slow rust way of bluing. The different steels used in the barrel material will blue or rust in different ways bringing out the pattern that the gunsmith forged into the steel.
The color case hardening parts are packed in a iron/steel box with bone meal, chared leather, the whole thing is brought to temperature (red heat) and then qunched in a water bath. The water usually is agitated with air, the air and the different materials in the packing box will gine the parts the color. The color case is not very deep nor is it very durable. After the parts are quenched cleaned/fitted to the gun they are coated with a protective coating like a varnish.
There is a section on gun making in the book "The Gun by W.W. Greener"
in the text he describes how damascus steel is made and how case hardnening is done.
To get the vivid case colors each gun maker have their own ways of doing the process. They do not share what they do.