You have the answer for paintings right there in your question. We take a small sample of the pigment that’s on the walls and that’s how we date the painting. The pigment is presumably not much older than the painting.
Carvings are much more complicated, and involve a variety of techniques. For example, stone weathers when it is exposed to the atmosphere. For various kinds of stone, for example, water from the atmosphere will penetrate into the stone and the depth of that water penetration will tell us how long ago the stone was exposed to the atmosphere. For a carving, that indicates when the stone was carved. There are other kinds of chemical reactions that happen for other kinds of stone, so this general idea can be used for a lot of commonly carved stones. Other strategies are used, including dating the carving based on its surroundings. This obviously is harder to do for things carved into walls, but imagine a stone object that has been carved which is buried in a heap of stuff that’s carbon-based. If this heap of stuff has been undisturbed since it was deposited, you can try to carbon date the stuff around the stone object and makes me reasonable assumption that all of that stuff was put down onto the ground at about the same time.
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