It is a proven fact that the gravitational force exerted upon an object directly affects that objects experience of time; the greater the gravity, the slower time passes, and visa versa. While the rate of radioactive decay may be nearly absolute, the gravitational environment in which that decay took place dictates how time was experienced by the decaying atom. Gravitational environment differs with respect to altitude, and would also differ based on the weight and density of material in which an object was located. Furthermore, gentle gravitational waves ripple across the face of this planet every day, creating the weather of our atmosphere. It is also highly probable that the earth has experienced a myriad of intense gravitational events over its history, during unstable periods of solar radiation, and from the ripples of other catastrophic cosmic events. I am very curious to read any research in this area.
For all intents and purposes, the Earth represents one frame of reference, as @kwinkunks states. Therefore no effect.
Yes the gravitational field on top of a mountain is slightly less than at sea level. This is very small though. Variations in gravity over the Earth's surface typically vary at most by a few hundred milligals (ie. "one part in 10,000" order of magnitude).
I doubt you would be able to measure General Relativistic effects due to such small field changes (even integrated over 10,000 years) - and even if you could, they are much smaller than other sources of error (e.g. the atmospheric C14 curve, and sampling errors).
How do archaeologists address time dilation when analyzing carbon dating results?
They don't. There is NO point in doing so. Compare two hypothetical substances at the peak of Chimborazo (the highest peak in the world with respect to distance from the center of the Earth). Suppose one of those substances spent all of the last 4.5 billion years at several thousand meters above sea level, while the other spent almost all of the last 4.5 billion years at the center of the Earth before migrating to the top of Chimborazo. That's the very worst time dilation possible. The substance that spent almost all of its earthly existence at the center of the Earth will be a couple of days younger than the other substance.
That couple of days difference over a span of 4.5 billion years is the worst case. This is many orders of magnitude smaller than the statistically significant experimental errors. The error induced by relativistic effects with respect to carbon dating is unmeasurably small.
Also please note the important difference between Gravity waves which come from hydrodynamics, and which you apparently refer to, when saying "gravitational waves ripple across the face of this planet every day".
This is ofc not true, as the waves that contribute to energy transport in the atmosphere are not the gravitational waves from general relativity.
Some astrophysicists would be really happy, if what you state were true, namely that gravitational waves can alter the decay time of atoms measurably. This is in fact not the case, why projects like LISA and other projects on earth have been set up to detect gravitational waves.
Also the proposed wavelengths of gravitational waves are so huge, that earth would only experience the wave near-uniformly with differential effects (that lead to measurability) being ever smaller.