Looking at the USA, it would be far more likely that American lunar colonies would be given a path to statehood long before independence came up. The USA already has two states not on the American mainland. The only colony rejected from being integrated was the Philippines and that was because Americans didn’t want to integrate such a large non-white population into the country. I don’t think a lunar colony colonized by Americans would have that issue.

Looking at China, I don’t know. China would likely attempt a similar process of integration, maybe to the point where senior political leaders have a stint running a bureaucracy on a lunar colony.

As to whether the Moon would unify its colonies as they kick out Earth colonizers, likely not. The independence wave in the Americas never united across across colonizing powers. The only way that there would be a united Moon is if one nation conquered the Moon and lost it or post colonization wars saw a post-colonization country conquer the Moon; this would likely not happen given the Moon’s strategic importance to Earth. Earth could accept an independent Mars; Earth could never accept an independent and unified Moon.