It's all quite confusing.
The 9mm P is a high-power, high-energy cartridge. Putting it in a blowback gun neccessitaes a very heavy slide, as the mass of the slide, and it's inertia, are all that hold the breech shut while the bullet is in the barrel. If you put the 9mm P in a blowback with a light slide, it would open too fast, and release high-pressure gasses into the gun and shooter's face. Not good.
With a locked breech, the barrel is physically locked to the slide for the time it takes for gas pressure to drop to a reasonable level. Recoil energy from accelerating the bullet is used to impart momentum to the locked-together assembly, which is used to unlock the assembly and eject the case, while stored energy from a spring is used to reload and relock it.
Blowbacks technically are gas operated, as the combustion gas pressure against the cartridge and the breech face is used to impart momentum to the slide, not the recoil energy of the bullet.
All of the modern mini-guns are locked-breech. Even the Kel-Tec .32 is a locked-breech auto. The locked operation allows the rest of the gun to be built lighter, as the high energy levels are directed down the barrel. In the case of locked breeches on the .32 Kel-Tec, and the Colt Govt. Model .380, it allows for a REALLY small gun to be built, as the frame doesn'y need to be stout or heavy enough to withstand the impact of a fast-moving slide, and with the lesser slide velocity, the felt recoil is minimal.
There are exceptions to the locked/blowback rules. One was the Detonics Pocket Nine, a gun well known for it's severe recoil. Another is the Astra 400/600 series, blowback guns which are beefy, heavy, and with hammer and recoil springs so stiff they make it difficult to rack the slide. Astras, however, are big enough to not have prohibitive recoil.
There are compromise systems available, just to add to the confusion. Gas-delayed blowbacks like the HK P7 and the Heritage Stealth, or the obscure Steyr GB, use gasses tapped from the barrel to keep the slide from moving until the pressure has dropped enough by providing pressure against a piston, or the inside face of the slide, in the case of the GB. Other than the piston, there's nothing locking thebarrel to the slide in these guns, and they can be chambered in high-energy rounds, as the higher the gas pressure, the harder the slide is held shut against the recoil impulse. An imbalance between the size of the gas port in the barrel, the surface area that the gasses work against, and the amount of pressure on the breechface allows the mechanism to function, as the breechface is set up as a bit bigger surface to work against.
Desert Eagles are are gas operated mostly because they have the space to build such a system in the HUGE platform the gun represents. They're set-up that way because the heavy magnum rounds have a great deal of energy to contend with, making recoil operation problematic, as there's just a bit much to deal with. There ARE recoil operated heavy mags, though. The LAR Grizzly is a scaled-up 1911, and the Automag IV's and V's are Browning-style linkless tilting barrel set-ups. Coonan .357's are another example.
Not having a schematic in front of me, I don't recall the exact nature of the D.E. gas system, but I THINK it's direct-impingement, like an FN-FAL.. Don't hold me to that, I could be wrong. The memory is spotty, and I don't own one. They DO have a gas tube running under the barrel, with a port near the muzzle. You must never run un-jacketed bullets in your D.E., because you can clog the gas port, and turn your gun into a single shot that can only be repaired by the factory.
Some .308 machine guns ARE recoil operated. The Maxim gun comes to mind. Sub-machine guns are typically blowback, as a sub-machine gun is big enough that a heavy slide (Or bolt.) is not a detriment to handling characteristics like it would be on a handgun. Early Thompson sub-machine guns had a locking system, (The Blish lock. It operates on the differing friction coefficeints of different metals under pressure.) later guns were simple blowbacks as the locking system proved uneccessary, and it was simpler (Read: cheaper and faster.)to make the guns without it.
I hope this helps. Did I miss anything?