Use stiff, but not hard bristle brushes for cementing. Nylon scrubbing brushes have a good stiffness without being too hard. Some natural bristle brushes are very hard and scratch the came excessively. In general, moderately stiff brushes with about 1 1/2" bristles are fine for cementing. As they do not last very long, they should be cheap, but with firmly attached bristle bunches.
Cleaning the brushes is very simple. The action of rubbing the cement under the leads with whiting causes a natural cleaning action to take place. As the bristles flex back and forward over the came, the cement is forced upward toward the handle, and then outward between the bristle bunches. Only a little effort is required to finish the cleaning: push a rounded stick between the bunches to move out the remaining cement. You now have a clean brush for the next job.
The alternative is keeping the brush in water, but this presents the problem of getting rid of the water (oil and water do not mix) before beginning to cement. As the water will emulsify with the linseed oil, it will be carried into the putty, leaving gaps in the cement when the water eventually evaporates. The cement will eventually harden, even though in water, as linseed oil cures by creating an organic polymer through oxidisation. It can also rot the wood handles.
Keeping the brush in mineral spirits does keep the brush flexible but requires drying/evaporating the spirit before beginning the cementing to avoid the residue of the spirit creating cement that is too thin at the start. This can be a really messy problem!
If you choose the “dry” method, it is important to keep the brushes free of hardened cement as it will scratch the leads badly, if not the glass also. Most brushes will only last 5-10 uses, and as they are not expensive, should be easy to throw away.