We've been discussing this together, and are a bit uncertain how to answer this question, and how much depth you are requiring. I'm going to post something fairly basic, and if you need clarification, please ask.

Basically, there is no sharp line between obsolete and modern. Rather, there were a series of gradual improvements over a period of many centuries. The earliest muskets were heavy, crude, weapons in which a lit piece of slowmatch was touched to a small amount of blackpowder held in a pan - matchlocks. They were not perfectly reliable, but reliable enough to be fielded by large armies to the exclusion of earlier missile weapons. If they hadn't worked reasonably well, they simply would not have been adopted. These were replaced by first the wheellock (16th century), and then the flintlock (17th century), weapons which required no match, but by means of friction, produced sparks to ignite the powder. These were finally refined into the percussion cap musket (late 1830s through 1860s), which were small copper caps containing chemical compounds that burned intensely when crushed. This, along with the development of machine-rifled barrels, would be the last major improvement to musketry; after that came the age of breechloaders.

The revolvers you reference were, prior to about 1872-73, almost universally of the cap-and-ball variety. They utilized a cylinder, in which several chambers were bored, containing the charge. When the hammer was cocked back, the cylinder rotated, thus exposing the next charge. The chambers themselves were loaded with either loose powder and ball or pre-made paper cartridges, tamped home with a rod located under the barrel, and then a percussion cap was applied to the nipple. The hammer struck the cap, the cap burned very intensely, igniting the powder and driving the bullet down the barrel.

This is not very different than the muskets produced from the 1830s through 1860s, with the notable exception that the revolver held six shots to the musket's one. Both used the same percussion lock system of ignition, and both were loaded with paper cartridges, though, of course, the musket was loaded from the muzzle. Certainly, reliability would not have been noticeably different. By virtue of using a percussion lock instead of a flintlock, reliability was already improved; the percussion cap doesn't get wet like gunpowder does, so all you have to worry about is keeping the charge in the barrel dry. But even flintlocks, except in heavy downpours, were not inefficient. So long as a musket was in good condition, with fresh flint and properly hardened frizzen, kept clean and dry, it would fire damned near every time. Of course, that's a matter easier said than done, especially in a military context.

Around 1872-73, Smith and Wesson’s patent on the design having expired, metallic cartridge revolvers began to flood the marketplace; rifles using metallic cartridges had begun to appear in the late 1860s, but they used relatively crude rimfire cartridges of inferior power, and didn‘t become obviously superior to earlier paper cartridge breechloaders until the early 1870s, with the proliferation of the centerfire cartridge. The big thing that metallic cartridges brought to the table is that they were self-contained. The primer, located at the base of the cartridge, performs a function very similar to the cap. The powder is seated behind the bullet, and is nearly watertight. The powder is the same black powder used in muskets for centuries - it wouldn’t be until the turn of the 20th century that smokeless powders began to replace it. But there is no need to load in steps - you merely insert the cartridge into a chamber, cock a hammer, and fire

What I am saying to you is that the closest we can get to a definitive change from antique to modern is, possibly, the switch from weapons loaded with loose powder and shot or paper cartridge, to weapons loaded with metallic cartridges. But even that should be viewed as an evolutionary, not a revolutionary, step.

So you are wondering about the advent of breach loading cartridge arms?