I am curious, can you flush the gas system with solvent?
Soak? yes, by submerging the entire barrel assy in solvent and waiting...
Flush? I don't see how, unless you rigged up something to pump the solvent into the gas system.
Find a picture of a Desert Eagle barrel. Just the barrel, it will help understanding it a bit. The large mass of the barrel assy at the muzzle is the gas cylinder. I say "barrel assy" because that's what DE calls it.
I have no idea how DE makes it, it boggles my mind, but however they do it, when its done, it is one solid piece. It does not come apart.
Gas is ported from the bore a short distance in front of the chamber, and flows though a channel (tube) underneath the bore to the front of the barrel where it does a 180 turn in the gas cylinder part of the barrel assy to push back against the gas piston which is mounted on the forward end of the recoil springs guide, and pushes the slide back.
Unlike the AR or the M1 /M14 sytems, there is no separate gas tube, gas block, or gas cylinder that can be removed. Its all internal to the one piece barrel assy.
Powder gas residue isn't the problem (DE comes with a "reamer" tool for the gas cylinder if it does) Copper isn't the problem. Lead and particularly lead bullet lube is the problem. The gas tube it small, it cannot be scrubbed so if it gets clogged with something that doesn't EASILY dissolve, you are pretty much SOL.
The factory won't fix it. They will replace it. And bill you for it. It's not a covered repair. They are very clear telling you this, if you don't listen, you will pay, or you'll have a spendy (and awkward) manually operated repeater.
I like my Desert Eagles, there are a few things I think they could do better, (adjustable sights, slimming the grip a bit, etc,) but they are not revolvers, they are not recoil operated semis, and they don't have an adjustable gas system like my Wildey. They require a very specific diet, and if you feed them what revolvers and recoil operated semis gobbble up, the Desert Eagle will get sick and refuse to fly, in short order.
They also have to be held a certain way (the mag must "float) or they choke and jam. They are made to run on full house ammo (and only jacketed bullets) anything less than that and they don't cycle reliably, if at all.
Shortly after getting my first DE (.357) I shot a .38 out of it, to see what it would do. Seemed to work fine, but the second "shot" was only a "click".
I opened the gun and it ejected a fired .38 case. What had happened was the .38 had moved the slide back just enough to recock the hammer, but not eject, and the gun shut on the fired case. That was the end of my experiments with lighter loads in the DE.