semperfi - the fast shooters the 'see the bird-shoot the bird' win at pigeons. 'Back in the day' many of the best shooters that had migrated from trap to pigeons - and were top doubles shooters like Sears, Gravestock, Robinson - were used to shoot a first bird rising and a secon bird 'not so much'
Having said that - virtually every pigeon climbs, especially a strong one. The weaker bird tends to drop a little but if a weak bird its usually a LOT easier second shot. A couple of inches Doesn't/Shouldn't matter - unless you/me have mis judged or mis pointed enough to move the killing pattern off the bird and it slips into a fringe hole in the pattern. For pigens, trap, international trap I want to 'err' on the high side. Using these guns in Sporting is challenging sometimes for the dropping birds.
I know (or knew) a lot of the top pigeon shooters in the 60's through 90's and could say the following general things about O/U. The SxS usually had either bent barrels (Billy Purdue's Purdy and fabbri)or very high stocks to get the patterns above the front bead.
1.) first barrel was high. Second barrel was variable impact among the group below but very few shot at same point of impact. I can say that for certain for several top shooters as I once had one of Materelli's MX8's, Homer Clark's So3EELL when he quit shooting it in the 90's and one of Richard Smith's Brownings. John Tirelli once owned the SHO I bought from Buckhorn.
2.) all except one or two shooters (and shotguns)were very fast on the first shot. Among the ones that come to mind are Homer Clark, Richard Smith, Britt Robinson, Gene Sears,Rudy Etchen, Joe Devers, Ennio Materelli, and John Tirelli to name a few) - not many of their birds had a chance to 'drop' out of the Barnaby trap and all birds have to jump out of Euro style fall away traps as there is no 'sling' like a Barnaby.
3.)Those that shot Columbaire as well as Box birds used the same gun - some may change choke or second barrel and many box bird shooter that shot trap also shot the same gun for box pigeons and guys like Robinson, Sears, Devers were killers at all the yardages and types of Pigeons - with their 'one gun does all'.
4.) For Barnaby traps it is easier to shoot too low than high, for Euro 'fall away' traps - just the opposite. I've seen the spring/hammock set in Barnaby's that threw the pigeon 4-5 feet in the air - which is a bitch for a flat shooting gun, but otherwise the bird is usually making its move within a foot or two. I hold low right on the center trap for the first bird so I am moving up for the first shot but I still want a slightly high first barrel.
For me, a target I REAALY had to concentrate on was the second shot on a fast crossing grass skimmer. It is all to easy to suddenly lose focus on the bird's head when there is so much background grass - shoot behind... but for a high shooting second barrel its too much (for me) to also have to hold under a bird that may suddenly climb to get over the fence.
What worked for me may not work for you - that may explain why there are son many different types for pigeon shooters.
I can also say that there was a tremendous migration from Brownings to Perazzi's and the sidelock market for Pigeon Shooters went from nearly always SO3EELL to a mix of Fabbri and Beretta. The 'kids' that reminded me of me that came to the sport with some ability, lots of ambition but few discretionary dollars (i.e. not the 1%) had to beg, borrow or steal to go from Browning to Perazzi. Today many of them will shoot Beretta 680 series or Browning 325/425 or an old used Broadway but many also will get their first used MX8 or Mirage 30 or 32 inch barrel for box flyers.
I just checked - I have owned 23 Perazzi's (some for re-sale) since 1969, boxlock and sidelock - Standard, SC3, SCO and SHO. I patterned at least 10, those that I intended to keep, 'if', and all but two shot bottom barrel higher than top. None shot same POI for me.
The two that shot lower had high stocks to 'raise' the POI but I don't like to see much rib - so I got rid of them. The three that are still in my hands are 30" SC3 and SCO's. My 'former Marine' sons have two each of my remaining collection. They all shoot with noticably higher bottom barrel POI.
That's just my experience being reflected here and you know what you are doing.
Regards,
Bill