There is, -or was,- a species of brush for the job. Mine is a Discwasher and I suppose it is English, from the spelling.
It is a block of wood (looks like Teak) with a Moquette-type fabric affixed. The fabric is biased at about 45degrees, not directly up from the material. It is used 'wet' with cleaning solution (long gone. I use isopropyl now) and with the 'bristles' scrubbing into the groove.

The same sort of 'brush' is used to remove lint from clothing.

It was expensive when I bought it.

A sound expert told me long ago that you could also run the stylus in a groove wet with isopropyl, which cuts out some of the dust-crackle.
I've done it, using an artist's paintbrush. Tedious, but it works.