Long term composition of matter
To answer your question, we can use the idea of radioactive decay;

According to the IB Physics Book from Oxford University, radioactive decay is "a naturally occurring process in which the nucleus of an unstable atom will spontaneously change into a different nuclear configuration by the emission of combinations of alpha particles, beta particles, and gamma radiation. Alpha particles are simply a Helium atom while beta particles consist of either electrons or positrons (the antiparticle of the electron).

Radioactive decay is a random process and therefore it is impossible to predict which exact nucleus may decay next. However, with a large sample of nuclei, say of your igneous rock, it is highly probable that in a given amount of time, a predictable amount of nuclei will decay, even if we cannot predict which ones exactly. Hence we have the idea of half-life, which is the amount of time it takes for the half of the initial nuclei of a sample to decay.


  approximately what percentage of the atoms (or protons, etc) in the rock might be the same ones that were in it 10,000 years ago?


It depends on what the igneous rock consists of; if it is made out of ununoctium-294, an infinitesimal amount of it will remain, as the half-life of it is 5 milliseconds. On the other hand, if the rock consists of Uranium-238, after 10 thousand years the composition of the atomic composition of the rock will have barely changed, as Uranium-238 has a half-life of 4500000000 years. So to answer your question, you would need to look at the half-life of your igneous rock. For example, if your igneous rock has a half life of 10 thousand years, then only 50% of the nuclei would be what they were 10 thousand years ago. However, it would also depend on how much of the rock is radioactive, as MaxW mentioned.


  Is it reasonable to even think of a "particular", individual atom?


As already mentioned, it is difficult to predict which particular nucleus will decay next, therefore, looking at the substance as whole facilitates predicting how much will decay in a certain period of time.


  Is it the "formation" of particles that persists over time, or is it the actual "matter"?


I do not quite understand this question. Nonetheless, I would say that it is the matter that persists over time, as the matter consists of these particles. In the photo provided below, you can see that the nuclei are decaying, but they are not decaying row for row, yet more in a random manner. Hence, I would not consider it to be the formation of particles that consists over time, but the matter as a whole.