Without knowing exactly how the maker finished the instrument (oil, varnish, shellac, wax, nothing, etc.) it’s impossible to recommend the “best” method.
That said, and the warning to check with the maker first applying, the following agents can be used on Boxwood - by that I mean it won’t hurt the wood, however some of these will remove certain finishes: Soap/water solution, paint thinner, denatured alcohol, acetone. Any and all of these “may” be enough, and any or all of these “may” leech the color out of the wood temporarily, but oiling normally restores the color. However, very often the problem lies not on the surface, but deeper in the wood. I don’t have time to describe in complete detail, but essentially, the final surface finish has a lot to do with the how the instrument was turned - the sharpness of the cutting tool, the speed at which the wood was turned etc., all of which affect the smoothness of the finish on a microscopic level. It is NOT just a function of what final grit sandpaper was used, because during the final turning process, a certain amount of tearing out of wood fibers below the surface occurs, the more this happens, the more common it is for dirt to accumlate below the surface, and then one can usually only get to the dirt by sanding. Sanding is fine, when done properly, however this will change the color of the wood, which many find an undesirable side effect.
At any rate, the differences in why one Boxwood instrument resists getting dirty more than another can have to do with the way the maker turns and sands, or it may have to do with the final wax, shellac, varnish, etc. finish, or both. I won’t speculate with regards to Mr. Olwell’s work (how he approaches finishing), as someone who works for him posts here on occasion and may or may not choose to give an answer, certainly it’s not for me to say.
One last important warning about using chemicals to clean one’s flute: Real Ivory can withstand all the chemicals I’ve listed, however, many of the Ivory subsitutes used to make the sort of rings found on Boxwood and other flutes, will not stand up to these same chemicals (or others), many Ivory substitutes can, and will, melt and or deform. My suggestion is that you NOT use ANY chemicals anywhere near these imitation rings, unless the maker of your flute has assured you it is safe.
Loren