I've shot a number of top break revolvers over the years, but all were antiques. My goal when shooting them was mainly to do no harm, which isn't necessarily the same as using something right.
Howdy
Actually, trying to do no harm isn't a bad place to start. All of my experience with Top Breaks is with antiques. If I owned an Uberti reproduction I think I would shoot it the same way.
These guns were designed so that when the barrel was swung all the way down, the extractor would pop down, theoretically throwing all the empties clear. I have found it doesn't always work that way, and an empty or two can indeed become lodged under the extractor star. You have to give the barrel a pretty healthy snap down for all the empties to fall clear, and I don't like to snap my antique barrels down that forcefully. I have found that by using gravity to help there is much less chance of an empty becoming lodged under the extractor. Sometimes I shoot a Top Break in Cowboy Action Shooting. We always unload at an unloading table. I stand at the unloading table and while I am rotating the barrel down I also rotate the gun in my hands so that it is parallel with the ground. In fact the barrel is pointing up just a tad. That way, the extractor pushes the empties out and when it snaps back they fall onto the table. Yes, to be clear, the barrel is pointing directly at me when I do this, but it is incapable of firing in this condition, and besides I have just emptied it.
Think of all those times you cringe when an actor on TV or in a movie slams a side swing revolver shut by flicking his wrist. Just don't do that kind of stuff. Open the revolver firmly, but don't slam the barrel all the way down until it stops against the frame. When the barrel stops against the frame it should do so firmly but gently. No slamming.
Same with closing the gun again. Some Top Breaks like the New Model Number Three require that the hammer be back in the 'safety cock' position so the hammer can clear the latch. Regardless, when I am closing a Top Break I index the cylinder with my thumb and forefinger in two flutes on either side of the top strap so a chamber is lined up with the bore. Then I close the barrel firmly but gently, making sure the latch pops all the way shut.
Just treat them gently, don't abuse them. Think of that dumb actor who knows nothing about guns slamming a side swing revolver shut and do the opposite.