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Forgive me if this has been asked, but what does an AGR do for a tuba?

I had one on a Yorkbrunner; couldn't tell the difference. Maybe I'm a putz!

I have one on my Miraphone 1291 5V BBb. Mechanically... it changes the distance between the end of the mouthpiece shank and the beginning of the leadpipe. Acoustically... I'm sure it makes some changes in the response of the horn. But, I really haven't spent much time fooling with it.

Matt Walters (Dillon Music) can offer you a much better explanation. Maybe he'll chime in.

I've never tried it on a tuba, but I have tried it on euphonium. The Adams Custom has that feature, and I spent a good bit of time playing one at ITEC 2010. I was having trouble playing one very difficult piece - missing long jumps to high notes. The salesman came over and adjusted the gap wider. Then the passage was easier to play accurately - noticeably. The original setting, at its narrowest gap, give slightly quicker-feeling response, and the wider gap gave a less immediate response and a generally "fuller" feeling (I'm not even sure what I mean by that; it's just the word that comes to mind).

There is also a subtle change in tone, moving from a teeny bit brighter at the narrowest gap to a teeny bit darker at the wider gap settings.

BTW, as far as I can tell, the Adams Custom tuba does not have this feature:
http://www.adams-music.com/nl/wf/instruments/tuba/