Today I put together a simple, pull-through cleaning kit to go in the cavity I carved out of the butt.
The pull-through is made from a piece of tarred bank line about 8" longer than the barrel. I tied a loop with half hitches on one end to hold a cleaning patch, while the other end is tied to a Countycomm Peanut Lighter. This gives me a weight to drop the line through the bore and it has a secondary purpose. It has enough weight that it should be able to knock some snow or mud out of the muzzle in the event you drop the gun.
The Peanut Lighter is about a half inch in diameter so it'll fit in a 20 gauge or larger barrel. It might fit through a 28 gauge barrel, depending on the choke. If you want to make a pull-through for a .410 shotgun you'll need to use something else, such as a length of brass rod with a hole drilled in it for the line.
You could use paracord but it will take up more space. The storage cavity in my stock is small so the tarred bank line, or even mason's twine, work better for me.
A few cotton flannel cleaning patches, a sample tube of gun oil from Brownells, and a scraper from Countycomm complete the kit, which is enclosed in a small Ziploc bag. The kit is a snug fit in the stock cavity and held in place by the slip-on recoil pad.
This is far from a complete cleaning kit but it'll allow you to do some basic maintenance in the field.