model of steam-era machine shop

[Moby Duck]

There is some interesting information on these "rings" on
Line shaft Mice

where it is quoted,
"They are line shaft wipers or commonly called mice as they scamper along the shaft."
The purpose as I understand it is to remove the chance of a catastrophic failure. Should a belt fall off a pulley and wrap around the line shaft one of two things will happen, the line shaft will get pulled down or the thing on the other end will be pulled up. The mice keep the shaft clean and polished, therefore "friction-less" or slippery and much less likely to grip the belt. As the oldtimer says "never put a shaft up unless it is polished"."

That makes sense to me. They also distribute the oil leakage from the bearings along the shaft to help with the slippery effect.
Kids all had these on their bikes in the 40's and 50's to keep the hubs clean. Usually made from leather or an old rubber tube, carefully cut with a barbed arrow at one end and a slit at the other. Making these mice was often a lesson in how not to cut yourself badly again. They do not actually need to rotate and seem to do the same job even if pendulous.