The Age of Exploration has ended so long ago that we have all collectively forgotten how it works. Almost every square meter of land on Earth is covered by one group or another of People with Guns. The only ones that aren't are ones so deeply undesirable that nobody cares.
But it actually works in a fairly intuitive manner. People who get to the Moon can do as they like there. Anyone who wants to stop them is invited to go to the Moon and try. This does not mean immediately by shooting them. Talking is a classic option for many good reasons that won't be going anywhere. But if you can't get there at all, you have no stake. If you can't get to the moon, you will have no say.
On Earth it is easy to forget that enforcement of laws requires enforcers to be able to get to the location where it will be enforced, because humanity conquered the task of getting from anywhere people are to anywhere people are at a reasonable speed a while ago. But this is still a base requirement. It hasn't gone anywhere. It has just been a long time since it was in question and people forget about it.
Space exploration will modify this for some period of time because in the Age of Exploration, people could live on the land. While people living in space require an Earth footprint to maintain themselves, that will give people on Earth a mechanism for enforcing power over the moon based on that footprint. But science fiction is full unto overflowing of space colonies going to war with Earth once they are capable of being independent for a reason; it's not so much a "trope" as a plainly obvious consequence of independence. Better treatments observe that Earth will know this and take it into account as it tries to hamstring its colonies from being independent, but A: this will eventually take the form of overtly hostile acts if space colonization continues on that will provide abundant casus belli for the colonies and B: there's a huge economic incentive to make them as independent as possible because it will always be more expensive to supply space colonies from Earth than to supply them from space resources.
Getting back to the original question, what will happen is this: Anyone who can get to the moon and stay there will have a say. Anyone who can't won't. Wringing hands, abtruse philosophical arguments, having hissy fits, screaming really loudly about "rights", complaining, none of it will matter. "Power comes from the barrel of a gun" is actually downstream of "power comes from being there". If you can't be there, you will have no power.
Also, I'm not particularly celebrating or condemning this in this post. This is just how it is.