It does seem difficult to avoid the conclusion that this world must be a single political unit with oddly defined member states. If you dig in on each country being a fully independent state I can't think of any good reason people would naturally come to regard borders in this fashion, or even continue to regard borders in this fashion for any considerable length of time if they were simply assumed to regard borders in this fashion by hypothesis. Interstate conflict, if permitted, would soon disrupt border definitions. So, to maintain the stability of the concept of countryhood, interstate conflict must somehow be suppressed, likely by an overarching world governmental authority. Then the story would be that this is just how the world government organizes itself, community laws in a spot being governed by those geographically in the community of that spot. Countries are the top level communities of radius 300 km. Perhaps there are also lower-level communities with smaller radii that define laws or ordinances within the sphere of authority determined by the overarching authority. (For example, countries might legislate the definition of violent crimes, but obscenity law could be the purview of "cities", communities of radius 30 km.) The effect would be a spectrum shift in law as you move from one place to another. A short drive might offer a bounty of legal pornographic filth, or freedom from the temptation of its open sale, but you'll be driving for quite a while to find a country where the knockout game is a felony instead of the national sport or vice versa.
Having leaders for an undefined number of countries would be tricky, so I think the main two options are direct democracy (with a location's laws being determined by the past votes of voters within the community radius) and autocracy (everyone is dictator of beneath their own feet for life!) The autocratic option is only stable if the power of countries is very, very strictly limited, as it is difficult to see how the countries would not be in constant conflict.
Still, I am intrigued by the question as you posed it, and I think we can get an interesting picture by borging up the population a bit. 274
Give everyone a neural implant that implements the geographic direct democracy in real time. If your implant can only communicate with implants within 300km, 300km is the radius of your country. Interestingly, if we can rule out long range missiles and remote drones, this also eliminates the possibility of international conflict without the need for an overarching supernational authority. If you have to get within 300 km of someone/something to attack them/it, there simply are no available international targets. Sending a bomber or soldier into rival territory can only be a transfer of citizenship and equipment. International relations are effectively limited to trade (by relay or immigration) and teleconference.
I don't think immigration controls as borders could be a thing, everyone would wind up effectively frozen in place (you don't have to move very far to cross SOME border), but you might wind up with prices of entry to move a particular direction. "If you're coming this way, bring copper." "If you're coming this way, bring food." With the neural implant direct democracy, this would be easy. People in food impoverished countries know they're hungry, and their neural implants vote to not allow people in without food. (Hopefully you can check the laws of a destination before departing even if your neural implant cannot directly communicate with the implants there.) Global trade would become very efficient. 207