My wife has a really nice pair of sandals (Munro Shoes). The soles and small heel are made from black, latex natural rubber. About six years old. They are in good to very good condition.
Problem: There is a white dust “bloom” on the rubber. A damp cloth will clean the bloom but once they air dry the bloom begins to return. We were thinking a slight application of baby oil might work but simple online searches reveals not to use oil on latex natural rubber.
Bloom is pretty typical of natural rubber compounds. It is the chemicals coming to the surface. In the rubber where I work, it’s usually zinc stearate…
You may be able to wipe a silicone solution on the rubber (like a silicone spray)and have it go away for a while. Even if you do that, it will eventually come back.
I have the same problems with the natural rubbers I work with. Sometimes they aren’t cured out enough to keep the chemicals in the rubber. And sometimes it’s just the mixing of the rubber.
And yes, natural rubber will absorb oil. If it absorbs enough oil it will turn into a gooey mess. If you try to heat it, you could revert the rubber and turn it into a gooey mess.
Armor All or related plastic / rubber penetrant dressings will get rid of the dusty look on the natural rubber for a few weeks to a month or two and give it’s original color back. Re-apply as needed. Make sure the dressing is applied to the rubber only and not the leather or cloth part of the shoe upper. Wetting a paper or small cloth towel with the dressing and carefully applying it works well.