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dear law & order
Look, I'm not an expert, but I'm pretty sure no country has an embassy in New York. (Consulates and all the diplomatic missions you want, why not. And if I'm wrong about the embassies, I'll say so.)
signed,
Washington, DC
you know, the actual capital.
signed,
Washington, DC
you know, the actual capital.

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(c) Consulates General. :-D Sorry. Tried to leave it, and couldn't.
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You're evil! Evil, I tell you! And all the passer bys would agree with me. And my brother-in-laws!
*g*
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There are a couple of exceptions I can think of, off the top of my head: Tel Aviv is not the capital of Israel, yet hosts a large number of embassies due to sensitivities regarding Jerusalem; and most foreign diplomatic missions to Myanmar/Burma remain in Rangoon/Yangon, rather than the new capital of Naypyidaw (and I don't believe there are plans to forcibly relocate the diplomatic missions--?).
But generally speaking, yes: embassies will be located in a country's capital. A consulate-general is a secondary branch of a foreign diplomatic mission; a High Commission is the mission of one Commonwealth country to another one. Permanent missions to the UN are not embassies -- even if a country's diplomatic relations are maintained through them. A diplomatic relationship is maintained through an accredited representative (an ambassador, or an honourary consul, or a visiting diplomat from an embassy located in a third country), but an embassy is an institution-- not only does it provide services (passports, consular, etc), but its physical grounds are considered native soil of that embassy's country. It's how people can claim political asylum by flinging themselves over embassy fences! You can't fling yourself through the office door of your UN permanent mission and expect to be granted asylum; you'd still be in US territory. So there are definitely important differences.
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its physical grounds are considered native soil of that embassy's country.
Interestingly, at least per wikipedia this is not precisely true. (It is true that people can seek sanctuary inside, though, which is your main point here and one with which I don't disagree.)
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