It is very dangerous to be pedantic about wall thickness measurements: I have successfully nitro re-proofed both steel and damascus tubes down to 0.014" MWT. However, I wouldn't recommend this as a hobby!
The thinnest part of a barrel tube is likely to be at about half way but again, it depends on what the original, or subsequent, barrel filer was trying to achieve in regards to finished weight and balance. Often this thin patch will be only partially round the tube: the area beside top and bottom ribs often show a marked thinning.
Beyond this half way point, the wall thickness will usually gradually increase towards the muzzle by up to 0.010-0.020". This is partly brought about by the choke constriction but even with cylinder chokes, I would expect to see a 0.010" increase in all but muzzleloader conversions.
I have always been lead to believe that this is to help protect the susceptible muzzle area from dent damage and also increase the swing momentum in lightly barreled guns.
Thickness at the other end of the tube is obviously of much more important in regards to safety and lightly barrelled guns with lengthened chambers are often considered a real hazard.
I have never seen nor heard of a barrel bursting due to this treatment but I will admit that it must increase the risk of a blockage burst.
As to the minimum thicknesses of the tube wall at 9", I personally wouldn't worry about it in a gun that has (near) original and sensibly tapered tubes from breech to muzzle and has not been lapped out beyond its nominal gauge range.
Put it another way, if the minimum wall thickness elsewhere is at 0.020" or over, the tube is internally parallel until the choke constriction and the barrel hasn't been seriously messed about with on the outside: Relax.
