If a country were to be annexed by another (for example, Latvia by Russia), what happens to the citizenship of Latvians abroad?
For example, do they keep their EU freedom of movement rights?
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The citizenship of the Latvians living abroad would be the same as Latvians at home. Their Latvian citizenship would continue to be recognised by EU, NATO and allied countries, but not by Russia and her allies.
An annexation would not be recognised by members of the EU. Latvia is a member of NATO and if it were invaded by Russia, it could call upon its NATO allies under the Section 5 responsibilities to defend its borders. Western countries would not recognise a de facto Russian occupation and treat the Latvian government in exile as the legitimate government, and so recognise the right of Latvian embassies to issue travel documents and so forth. It would be likely that Latvians could claim asylum if it would be dangerous to return. Of course, Latvians living in Russia, or Russia's allies would find their position more difficult.
An invasion of Latvia could lead ultimately to a full war between NATO and Russia. The citzenship of Latvians would then ultimately be decided by the result of that war.
To look to historical analogies: In 1975, Indonesia invaded and occupied East Timor. Citizens of East Timor living in Australia continued to have their citizenship recognized as Timorese, and many chose to claim asylum in Australia.
In the case of voluntary annexation (such as the ascension of Texas to the USA, or the union of Scotland and England), the process could be much more orderly. Here a generally recognised and legitimate government, following a constitutional procedures, asks a foreign government to take over. The two governments could arrange matters of citizenship, generally one would assume that any expats would have their citizenship updated to that of the incoming government.
To expand on the answer by James K, the recognition of other states is mostly a bilateral issue between states, and governed as much by realpolitik as by principle. There may be a point where the non-recognition of realities becomes pointless, but states can still try.
Recognition of citizenship follows the recognition of states.
So the questions would be if there is a government in exile which is accepted by the host government and which can credibly claim to represent the occupied population. Credibility is in the eye of the beholder.