Shooting technical expert
There can be two reasons for chequering filling-in like this. The most common is honest sweat, which leaves the fore-end just sticky enough to attract dust.
The sweat seems to glue the muck in place. The second reason is the application of oil to the chequering, which is not advised.
Damp oil also attracts dust, and the oil itself also sets in the grooves in the chequering.
The best way of rectifying the situation is a relatively vigorous brushing with a stiff-bristled brush, like a nailbrush.
Whatever you do, don’t use a wire brush. If the muck is reluctant to come out, a lick of white spirit can be helpful – particularly with old oil.