Long, long way from being an expert here....................but thought I would try a simple response in good faith.
I suspect one could ask that question a lot of places and get a lot of good answers. I too ventured onto the benchrest circles for a while, trying to pick up all I could about their methods and rifles. Sure is a lot to pick up there in benchrest land, I might add................
..............pretty heavy group in their methods and materials.
Some of what I slowly picked up there:
Most shooters will have a type of rifle holding stand for cleaning the barrel. Some of these stands are collaspable for carrying in the gunbox with all the rest of the stuff they use to shoot/load/clean at the range. One does not really need a Suburban to haul all the stuff; the box(s) get heavy to carry around. All of these cleaning stands will position the rifle with the barrel muzzle lower than the breech, mainly that the solvents run out of the barrel and not onto those expensive triggers and gum them up; also to keep the solvents out of the bedding the best they can. The stand's contact points for the forearm/buttstock will be padded.
All will have some type of bore guide to protect the throat during brushing. After the costs of the premium barrel that are like jewels, and the labor to chamber/breech up etc to micro tolerances, wearing out the best of the barrel with a bare rod is not what they think of first. Most guides will have a section of the bore guide to keep solvents out of the action/trigger during patching. Hence a lot of the boreguides are two pieced. One piece is(stays on) on the brush rod, and cannot be pulled over the brush for obivious reasons of trying to protect the throat. The second part is larger in which the brush section inserts for cleaniness during patching out. The bore guides are precision built to be concentric with the bore and action, some will have an o-ring to keep the solvents up front in the bore(somewhat bore diameter related), most are simple renditions of the case involved for good fit; some can be specificly tapered with the same outline of the case involved. A lot will simply be a friction fit, some have a bolt arm to hold the item in place(bolt gun specific). The No.1 short breech presents some challenges there that the bolt guns do not share.
A lot of folks use the Dewey coated rods, I like them myself; had one set for many years. I try to brace the rod during brushing with one hand on the towel covered buttstock to prevent overdue flexing etc. A proper patch size is a good thing, if the rod is flexing in a big "U" then the patch is a way too tight. Lots of good rods on the market today. Most will have a rod holder for keeping the two rods(brush and patch) clean. Of course, one can get by with just one rod; the two rods are for convience.
Solvents is probably a big subject all in all. Many good ones one market, just have to try them and see how they suit you. Many homebrew types are on the internet, many just buy a solvent off the shelf, for instance Butch's Boreshine is just one example. A lot of the older guys use the Hoppies #9 with/without the good stuff as a basic powder solvent, the bore dried out with clean patches and then an application of Sweets for the copper fouling. Sweets is a sticky ammonia based solvent, kinda? has a smell to it, and most shooters do not leave it in the bore a long time(time in the bore is kinda their own experience). After the sweets is pushed out, and the bore dried with clean patches, another patch or two of Hoppies, then dried out again ready to shoot. Some folks use gloves with the solvents if they are sensitive to that type of thing.
I think with the No.1 some things above are some harder to achieve, but doable with the short breech section. A proper cleaning station with the barrel's muzzle down. A two piece rod guide(one section on the brush rod, one in the breech for alignment/patching) that is fitted(simple rod type, o-ringged, or specifically tapered to the chamber), the total bore guide length a bit longer than the tang/safety area. Black or white delrin can be used to make the guides. A protective covering over the buttstock, whether the stock is wood, synthetic or whatever, some of the solvents one can choose are not too good on some finishes. A set of good rods, brush and patch, an example are the coated rods(my choice) by Dewey; other good rods abound today. A brush that is all bronze, some cheaper brushes have a steel twisted section, the better ones in my opinion are all bronze to include the twisty. A good solvent(s) of your choice used with discretion as to your particular needs. I like to dry the bore with a few clean patches before shooting, just to be sure the bore is free of anything I left in there. Again little things like bracing the rod during use with one hand/finger on the buttstock and a proper fitting patch will make life easier after a while of time on the line. Did I mention, somekind of tight fitting; well designed bore guide, to the chamber and to the brush rod.
As you know, there are lot of guys with a lot of ideas, experience on how to's; this is just one observation and practicality for my own needs. Probably no such thing as the best way, as the industry is always evolving and shooter's methodology. Lots of shooters? will tell you their way is the best, only you can tell that for yourself with actual time in grade(hard earned experience). I do think that simply running a bare rod with patch or brush thru the bore is selling oneself short though.
Sorry, if too wordy, but is not a simplistic subject. I will remove if it offends....................
