I was concerned by the mess too, meaning fine abrasive dust & getting under the carriage wipers. First I wipe off the way oil between head stock & carriage just as a precaution. Not bone dry but not glistening wet. Then I lay down some old terry towel face cloths accordion style across the bed ways so they don't overhang much to interfere with the rotating lead screw / power feed shaft. On relatively short travel jobs its easy to just use a t-shirt material with small magnet on either end to hold it in position. Then to remove just fold it into itself, vacuum & remember which side is up to replace. If its white is shows any fine debris quite well. I also put a bit of rag stuffing in the chuck just behind the part so dust cant migrate into the chuck mechanism or into the spindle.
In my case (model 4-stroke cylinder liner) there was only about 0.004" (internal diameter) to remove so not really a lot of material. I was more concerned by abrasive material coming off the wheel dressing operation. The wheel is only 0.75" diameter & I held a vacuum up to it. Do what you can to keep things clean. Now if you have a bigger wheel like a 3-4" doing OD grinding, everything grows proportionately. Much more wheel to dress, more part material to grind. I agree, a dedicated lathe or better yet cylindrical grinding lathe would be nice. Unfortunately with a regular lathe the spindle & tooling all have to be decently accurate & tight so its not like any junker would do. I've thought about a dedicated mini setup like using a knock-off Asian lathe especially for small parts & dedicated spindle motor, but I don't do enough of this to warrant the expense. In hindsight I think regular machining of (in this case cast iron) has the potential for as much or more damaging mess. I do essentially the same rag procedure with CI to keep the swarf & dust off the ways.
There is also a lot you can do with lapping (and often its required after grinding anyways). But its a messy, time consuming ritual & also requires specialized tools. Lapping anything more than 1-2 thou is getting into the inefficient range.