CARRY ON GENTLEMEN!
I've avoided any effort to research pattern welded rifle and pistol barrels, but for your interest
Charles Daly (Sauer) Drilling Diamond Regent Grade. Barrels marked "Fine Damascus."
It is my sense that the transition from pattern welded rifled tubes to Bessemer/Decarbonized/Plain Steel/'Rolled (Bessemer) Steel' and eventually to fluid steel occurred much earlier than it did in shotguns, and was driven by improvements in military weapons
Fire-Arms Manufacture 1880 U.S. Department of Interior, Census Office "The earliest use of decarbonized steel or gun-barrels is generally credited to the Remingtons, who made steel barrels for North & Savage, of Middletown, Connecticut, and for the Ames Manufacturing company, of Chicopee, Massachusetts, as early as 1846. It is also stated that some time about 1848 Thomas Warner, a the Whitneyville works, incurred so much loss in the skelp-welding of iron barrels that he voluntarily substituted steel drilled barrels in his contract, making them of decarbonized steel, which was believed by him to be a novel expedient. The use of soft cast-steel was begun at Harper's Ferry about 1849. After 1873, all small-arms barrels turned out at the national armory at Springfield were made of decarbonized steel(a barrel of which will endure twice as heavy a charge as a wrought-iron barrel), Bessemer steel being used until 1878, and afterward Siemens-Martin steel."
