My Neil Miller Cordovan strop was at a stage where hand rubbing was no longer enough to clean and it was showing signs of a build up in places, So, having seen a video where Maestro Livi recommends lighter fluid as a cleaning agent I looked around to see what might be available - it's been many years since I smoked and even longer since I had a refillable lighter.
I keep brake disc cleaner available for when I'm polishing blades and I've just tried this on a piece of tissue paper - a squirt of fluid and then wipe the strop three or four times in the same direction. The effect was amazing in terms of the dirt that came away - and because this stuff evaporates quickly the strop was dry again within a few seconds. This time when I hand rubbed the difference was really noticeable. Something to bear in mind.