This is a subject I'm very familiar with...
Over time many rubbers will degrade; the worst offenders seem to be the nasty "soft touch" coatings that can be found on myriad products despite the fact that they've been known for years to break down into a sticky mess. These may be worse because they're thin, or perhaps because they need to be applied in some way (dipped, sprayed etc) rather than moulded like solid rubber objects.
The presence of organic solvents such as oils and greases will hasten this degradation; my Brompton grips are a proper mess thanks to my use of Vaseline on my hands every morning before the commute.
There appear to be two ways to treat degradation. With "soft touch" coatings the best approach appears to use a mild solvent such as isopropanol to remove the stuff completely - note that these are often applied to plastic substrates so use of anything too aggressive (i.e. acetone) will likely damage what's beneath.
The second approach is to try and stabilise it a bit seems non-solvent liquids might help in this regard, as might inert powdered stuff. I've found gym chalk dries my grips out for a while, however since I'm constantly exposing them to a problematic substance the issue just continues ad-infinitum.