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While we are talking of some slight and minimal damage to the bore of the barrel here, the major damage from using a cord type pull through is actually at the muzzle of the barrel.
We all know that the Crown on the rifling at the muzzle is a very large factor in accuracy. Recessed and 11 degree crowns are popular with target shooters who demand the greatest accuracy possible. You will never see a serious target shooter using a cord type pull through but rather cleaning rods.
There are two problems here. The flexible cord itself picks up debris from the barrel when it is first put into the barrel. This can be a mixture of unburnt powder, glaze from firing, metals from jackets, primer residue, and such, and all of it, over a time, can imbed itself in the cord, whether it be an issue pull through or a bore snake. The average cleaning can never get all of this embedded material out, and the average shooter will simply not take the time and effort to do a thorough cleaning of the pull through or bore snake. This material acts as an abrasive with the cord as a host or carrier.
The second problem is the fact that it is almost impossible to pull through cleaning materials straight without the cord touching the crown and rifling at the muzzle at some time. If you think about the rifle butt resting on the ground, with the cord coming out the muzzle, and your weak hand grasping the muzzle end of the rifle while you attempt to pull straight back with your other hand, then you will see this for yourself. You have to keep the muzzle perfectly centered while you pull backwards. If the patch inside the bore hits a tight spot, then it binds up a bit, needing extra pressure to pull it through, but the muzzle will get out of line, so that the cord is rubbing a bit on the end of the rifling grooves, lands and crown. This embedded abrasive material acts like a grinding or cutting compound, and will eventually wear down the rifling or bore at the muzzle.
I have seen, many times, people trying to use a pull through with the cord coming out of the muzzle at up to a 30 degree angle or more. We see posts here every week that ask about how to crown a rifle barrel on a Milsurps rifle. While we tend to think about our rifles being cleaned under the best conditions, the average Soldier is not really that interested in doing a job like we would do. He just wants to get his bore clean enough that the person inspecting it will not chew his butt. And the wearing process is a gradual one, that in most cases, it takes the trained eye of an Armourer to detect, not some NCO holding the rifle unsteadily up in the air and looking down the bore with his Mark I eyeball. However even a small bit of wear in the wrong place can affect the accuracy of the rifle.
If you want an EXTREME example of this, clamp a steel bar in a vise. Put a bit of valve grinding compound on top, take a clean piece of string, and pull it around the bar using a shoe-shine motion for about 50 movements. Then, look at the groove it made.
Many Nations issued steel cleaning rods for their rifles, but they also issued muzzle protectors (eg. Russia) or cleaning bolts (Sweden.) The Americans had individual cleaning kits, but they also had Squad or Barracks cleaning kits with long one piece rods.
The biggest problem with any flexible pull through systems is not bore damage, but rather muzzle damage because of the inherent problem of trying to keep everything in line, and the ability of the cords to pick up abrasive materials.
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Same handle.
