First, in my many years I can say that as far as I know, there's no such thing as a "dry closet".
They all seem to be damp and will rust metal and mildew leather and canvas.
Virtually every rusted rifle I've seen was stored in a closet and/or a fleece lined gun case or cloth bag.
Second, how do you KNOW the ammo you got is really non-corrosive?
Not so much these days but in the recent past people got ammo made over seas, especially in former Soviet Bloc ammo factories in which the production staff were a little too likely to use up some old corrosive primers in the commercial production.
Even if the box is marked "Non-Corrosive" and it's advertised as such, ALL foreign made ammo should be suspect until positively proven otherwise.
What makes me think you got corrosive primed ammo is that the military rifles rusted, the Winchester didn't.
To test, clean the barrel out so it's dry.
Pull a bullet from the suspect ammo and dump the powder.
Get a piece of sheet steel and polish it off with sand cloth to a bright finish, then degrease with lacquer thinner, Acetone, or 91% alcohol.
Load the primed case in the rifle, hold the muzzle a inch or so away from the clean sheet steel and fire the primer.
Keep an eye on the steel for a day or so in normal humidity.
If the plate quickly develops red rust..... The ammo IS corrosive.
What to do now that you have pitting.......
Go to Brownell's and buy a one-piece stainless steel cleaning rod.
NOT brass, not aluminum, not a bore snake, and not a screw-together rod.
Dewey stainless one-piece rods are sort of the default brand for most shooters.
Buy a brass button tip jag.
Buy a good bronze bore brush for each caliber.
Buy a jar of JB Bore Paste, or JB Bore Bright.
Use the brush and bore solvent to make about 10 passes through the bore to remove the surface rust.
Use the paste with the rod and cotton patches to polish out the remaining rust.
This will not remove pitting....nothing can "remove" pitting, pitting is forever.
The paste will remove the surface rust and shine the bore as well as possible.
DO NOT try to shoot the rust out. That can cause a KA-BOOM.
To store the guns so they will never rust again.....
Clean the bore before storage.
For military rifles, use hot water or a black powder bore cleaner.
Either will dissolve any corrosive primer residue.
Note that no modern bore solvent will dissolve and remove corrosive residue.
Only water or something that contains water will do.
Even Hoppe's #9 will no longer remove corrosive residue since the EPA made them change the formula.
Once the bore and bolt face are cleaned with water, then clean the bore with bore solvent and patches to remove carbon and copper fouling.
For modern civilian non-corrosive ammo use a standard bore solvent.
Once the bore is cleaned, dry thoroughly with a couple of patches, then apply a rust preventing lube.
Good ones are CLP Breakfree and the above mentioned RIG grease.
Both are known for excellent protection.
Again at Brownell's buy a "Bluguard" bag for each rifle.
Other sites sell the Z-Corr brand bags.
These bags contain a chemical that drives out oxygen and moisture that causes rust and replaces them with a vapor that surrounds the metal with a barrier that totally prevents rust for at least 10 years, and as much as 20 years.
This is how the military now store firearms. Unlike the old greases, the rifle can be removed from the bag, the bore wiped out and the rifle put directly into use.
As long as the bag is properly sealed and not compromised by nibbling mice or nosy family the guns will be in perfect condition for years and years.
https://www.brownells.com/gunsmith-t...prod42749.aspx
These bags can prevent rust even in a nice "dry closet".