Well, someone was buying a lot of Boss, Purdey, Woodward etc "bests". Not sure how many of them were going elsewhere in Europe for the nobility and other rich shooters. But in the 1890's, there wasn't the amount of driven shooting going on anywhere else as there was in the UK. And the other European gunmaking nations were also producing very fine doubles of their own.
Comparing purpose-built target guns (which the British also made) to the "game guns" used for driven shooting doesn't work. Your average British 12 bore game gun was lighter than virtually all American 12's. Typical weight would have been around 6 1/2#. The only British pair I've ever owned weighed 6 1/4 with 28" barrels. My current Lancaster SLE, also 28", hits that same weight. So it's not quite like putting thousands of rounds through your typical trap or skeet gun, which will usually weigh at least 1 1/2# more. In terms of shells fired, I've gone through close to a flat of shells for 3 days of driven shooting in Scotland. And back in "the day", the big shoots featured more birds than you'll typically see on your average commercial shoot today.
Also worth pointing out that not all the guns used for driven were bests, nor were all the shooters the super rich. My British pair were Army & Navy boxlocks made by Scott, ordered by a retired British brigadier (brigadier general). Nice guns, but they would not even have qualified as boxlock bests.