Marrakai.
Excellant story on your 577 Greener.
Regarding the video featuring gunsmith Ron Webb shooting pigs with a a Red Label double rifle, it is one of my old videos and was shot in the NT back in the early 1990's during a trip I did with him.
The rifle was a 20 gauge blued receiver Red Label in 375 Flanged magnum.
A very nice gun which had been fitted with a full length top rib, had bushed strikers and had much trigger and safety alteration and also had been restocked including a cheekpiece.
A couple of other nice guns made a short appearance on that tape.
One was a sidelock Hollis in 450-400 3 inch and the other was a boxlock Gibbs 360 No2.

Regarding some of the Yank Red Label rifles shooting loose, I know some were done in 375 Belted mag and if they did come off the face maybe this could explain why, as they could possibly have been loaded to pressures that were too high in an attempt to get bolt action 375 mag velocities.
The Red Label is a very strong action and I think that so long as pressures are kept at a level suitable for doubles there should not be any problems with them shooting loose.

Incidently, I know the 444 sleeved shotgun thingy you're talking about and actually have an old photo of the very same gun as it was once offered to me for sale.
Not my cup of tea though.