One reason British exports to this country might have declined during the 1880's-90's could be the growth of the American gun industry. Prior to 1880, Parker and Lefever combined hadn't produced much more than 20,000 guns. 80's and 90's, along came Ithaca and Smith. And of course the tariff, when it came in, gave all the American companies a significant edge. (Sort of like why we don't buy ethanol from Brazil today.) In spite of the bad economy, the 90's were very good years for the American shotgun industry, which was in much more of a "fledgling" state in 1880 than was the British industry. And the Brits had a pretty good export market for their guns even as sales declined here in the States: the Empire. Military officers, colonial officials, colonists etc. Meanwhile, the higher end of the British gun market got a significant boost because the most important man in the UK--the future King Edward--spent a considerable amount of time shooting driven birds (when he wasn't chasing women) during his long wait to take the throne. That gave shooting and the "best gun" trade a huge shot in the arm, for which there was really no equivalent in this country.
As for the guns used by the best competitive shooters, they had their guns furnished by the companies that sponsored them. Eventually, many of those pros switched to pumps to win big shoots. Likely had to do with American gun companies making a lot of pumps and trying to sell them to American shooters because so and so won the Grand (or whatever) shooting one.