24k is going to be the easiest for someone not familiar with the process. It works very easily.
If you have a very wide band,,you can use two or more gold wires layed side by side in the groove to make fill width and make what appears to be a solid band.
This works best and easiest with 24k, but you can get away with it with lower k gold too if you're patient.
The groove is cut to width and depth (you don't need to go subway bed depth).
Dovetail punch the sides. Some engraves cut the 'undercut' w/a very narrow onglet point.
Then burr up the bottom of the cut. I use the same punch as I use for the undercut/dovetail punch if it's narrow enough. Sometimes I just handle it sideways in the bottom of the cut so it fits.
The idea is to just throw burr teeth upward for the gold to impale itself on as it's set into the groove.
Same idea in putting an inlay of a figure/animal ect in or on the gun.
..Don't stab in the burrs and throw them up so high that they exceed the height of the groove. If they do they will show as tiny steel specs in your gold inlay/line when polished out.
If you blue or brown the steel part,,those specs will color as well. If you do get one or two poking thru, they can be punched back down thru the finished surface and the tiny spot burnished over. I've even inlayed a speck of gold into the gold as a repair for my mis-deed.
On the wide band inlay where I'm using more than 1 strand of wire, I run the burrs in opposite directions so they hold better.
It doesn't take long and there's no set pattern of BPI (burrs/per/inch!) Just tiny teeth to snag that gold.
When setting the gold in place, start the multiple wires carefully in the groove and punch them inplace in one spot,,and not at the very tip of the bunch. Come onto the strands a fraction to better get a solid strike.
Any ragged end can be snipped clean with a flat graver or better,, carefully bunched up in the groove back toward the punched strike with a punch,,the gold is soft. Leave it there so you can line it up squarely with the other end if doing a cylinder coming around the other end. Or simply now punch it down and end it there if it's below the wood line far enough for you.
Continue to punch in the multiple wire band carefully. It's feed itself sort of, into the groove as you punch it down.
I sometimes punch just ahead of the last swage , something like Acer mentioned above.
That keeps the gold from getting out in front of you too far if you're punching too hard (but the teeth usually hold it back for you).
It's also a little trick if you have cut a portion of the groove a bit too wide for the amt of gold you have to fill it.
Punching ahead of the wide point will capture the gold at that point and then any gold wire inbetween the two swages can't extrude like it wants to do. It'll inchworn in shape as you swage it in place between the two punch swage marks, But that extra room you provided in the spacious groove will take that gold and look just fine for it.
Otherwise you may have a portion of the groove,,that wide portion, shy on gold filler and that leaves a depression in the gold surface there and often an open line on either side of the gold and the edge of the groove. Not a good look.
The swaging of the multiple wires of 24k gold together must be done so they sit side by side. You don't want one to be an excess and extruding to a thin flap over the top of another. That flap or flake will do just that and peel off when polished.
Done right, the multiple gold wires will cold swage together and not be seen as separate pieces. Instead they will appear as a single piece of gold.
I've done hundreds and hundreds of gold figures, animals and birds ect with this process as well as line inlay and never had a problem with inlays coming out.
It's nothing I came up with,,it's one of many methods of used by engravers since forever.
Lower k alloys can be inlayed just as well. But not easily with the above method.
The single piece of wire or sheet first annealed then punched/swaged down into the cut or groove is the usual method.
Don't expect lower k gold the work easily like 24k. Fit the piece or wire carefully to the groove or cut you are inlaying.
It'll work harden quite fast, but realize that even steel is routinely inlayed into steel with common engravers tools. So it's technique that gets you results, not brute strength, or some sort of magic.
If any one thing for inlaying 10, 14 and even 18k gold sheet and strips I've found helpful,,is to clamp one side or edge of the inlay in place . Then punch down/swage the opposite end into place.
Then clamp the punched in place side and swage the previously clamped edge down.
That holds the piece from springing upwards on you. Sometimes just a small clamp right in the center and work the edges down is all thats needed.
Wire for line inlay goes right into the groove w/o any helper clamps of any sort as long as the groove is the correct size,,is undercut right and the wire isn't too big in dia.