One thing to be aware of with the 1873s is that the cartridge overall length must be within a certain range or they will not feed. In fact, they can jam up requiring you to disassemble the magazine tube to clear it, if your rounds are too short. I do not know what that OAL range is for the rifles in .357, so I suggest doing some research to ensure your rounds will work in your gun.
Howdy
There is no cartridge stop on the carrier of any of the toggle link rifles, Henry, 1866, or 1873. Instead, the rear wall of the opening in the frame where the carrier rides serves as a cartridge stop. So the overall length of a cartridge on the carrier determines how much of the next round in the magazine protrudes onto the carrier. There is a bevel at the bottom of the floor of the carrier. The purpose of the bevel is to shove a protruding round back into the magazine so the carrier can pass by on its way up to eject the empty brass and feed a new round into the chamber. If too much of the next round in the magazine protrudes into the carrier space, the bevel will be too short to shove the round back and the carrier will jam against the protruding round.
Here is a photo looking down on the carrier of my Uberti 1873. The carrier is in the lower position, and there is a round on the carrier. You can see how the magazine spring has shoved the next round partially out, and also shoved the round on the carrier back against the wall of the frame. This rifle happens to be chambered for 44-40, but the theory is the same with a 357 Mag/38 Special 1873. My 44-40 round is just about the perfect length, only the rim of the next round, about .060, is protruding into the carrier space. The bevel will easily shove it back as the carrier rises.
Most 357 Mag Uberti owners who shoot 38 Specials try to keep their rounds to an Over All Length of 1.45 - 1.50. This length will still allow the bevel to shove the next round back into the magazine as the carrier rises. However, each rifle is different and you may find you need a cartridge OAL of 1.60 to prevent jams.
NOTE: Do not use round nosed or pointed bullets in a rifle with a tubular magazine. Flat Point Round Nosed bullets or Truncated Cone bullets are best. Semi-Wadcutters have a flat nose, but sometimes the shoulder will jam on the edge of the chamber as it feeds.
The cartridge second from the right in this photo is a 38 Special loaded with a 125 grain Truncated Cone bullet.