What are "sleeved" shotgun barrels and how does it affect the value of the gun assuming it was done properly?

I understand the old barrels are cut off about three inches ahead of the breech and reamed out so new tubes can be inserted, as into a monobloc. Sometimes small-calibre rifle barrels are inserted in existing shotgun barrels but that is called something else, like an insert, perhaps.

The first process has been done by respectable British firms. It's not as good as new, marketwise, but better than worn-out or pitted barrels.

Or older Damascus barrels have a smaller gage tube sleeved in them to strengthen them so they can be shot.

There is another type of "sleeved shotgun barrel" out there. They are used in "still target" Turkey shoots. The object is to put 1 piece of shot dead center of the "X" which is the intersection of one vertical and one horizontal line on a white cardboard target.
Around our area various gun clubs and local fire Depts. hold these shoots as a source of funds for the organization.
The "sleeve" in these guns is a tube, usually about 1.5" long that is taper bored to approximately 16 Ga. full choke dimensions, that is soldered into the muzzle of a 12 Ga. barrel. The logic being, the more shot you put in the card the better chance you have of "cutting center".

Even done right, it reduces the value from what it would be with original barrels. But if the original barrels have rusted through or burst or something, it's still a good way to save the gun.

There's sleeving and there's re-lining.

Re-lining is where the original barrel is kept intact but it is bored out. A full length tube only only some few thousandths of an inch is then pushed up it its full length. This is then chambered and, on a shot gun you are pretty much ready to go.

The man, Teague, that offered it here in the UK has stopped offering it. It has problems in that if the barrels are then dented they can't apparently be repaired? Or some other reason. Bt he's stopped offering it.

Other weapons re-lined are where an old shot out rifle is bored out and a new full length rifled tube is then pushed up its full length and chambered. Either back to original calibre or in some cases for example boring out an old BSA Martini originally in .22LR and re-lining to say .357 Magnum or .44 Magnum.

Sleeving is where the ribs are sweated off the old barrels and then the barrels are cut down to say four inches length. Leaving a stubby four inch barrel sawn-off shotgun in effect.

These stubs are then bored out from the front end and a stepped sleeving tube inserted. This is in effect how Beretta make their new guns. They start with that stubby four inch sawn-off shotgun from the outset.

The new sleeving tubes are then blended to the same outside contour of the four inch stubby, the ribs are relaid and the barrels blacked.

On a good job yu won't see the join. On a Beretta it is "disguised" by engraving over the joint. So the engraved ring on a Beretta isn't for decoration it is to hide the join!

Value wise?

Yes it de-values the gun. The market prefers original barrels or a complete new set of barrels being made for the gun. But a wise buyer at auction can use that to their advantage to beat the market.

Is it safe? Of course. Does it affect the balance of the gun? If done badly and the new part isn't struck off with skill to duplicate the original barrel contour then yes.