Handbag lining - dust (mould?) coming off?

[BobbyNW]

My other half has a bag. Lining inside should apparently be cotton. It’s a dark grey/black colour. A load of black almost dust type substance comes off on to your fingers if you touch it. It also rubs off on to anything you put on the bag.

two questions:

1. any idea what it is, and
2. how do we get rid of it?

Depends why the actual bag is made of.
I’d probably turn it inside out and give it a scrub with a damp, not sopping wet, scrubbing brush that has been dipped in a water and cleaner of some sort. DAB the excess off on a dry cloth then scrub the lining, one small area at a time, dry each area with a towel before moving on to the next one. Leave inside out on the line to dry.