In most national elections there is usually the option for voters to cast their ballot in foreign consulates. But as far as I can tell this is usually a mere formality as very few voters ever make use of that privilege and the "foreign" votes don't really affect anything.
But has there been an election where the foreign vote has had a deciding effect? Here "deciding" means that the vote margin among foreign votes is larger than (and opposite to) the margin of all in-country votes.
The election of the US President in 2000 swung on the result in Florida, which Bush won by a majority of 537 (in the official count)
The system in the US is for foreign citizens to vote by post. Consulate voting is possible, but is only recommended if the postal system is considered to be unreliable. The overseas votes tend to favour Republicans, boosted by overseas military voters who tend to lean to the right.
Had all the overseas votes been ignored, then Al Gore would have become president.
To say that overseas voters had deciding effect is a little unfair. You could equally say that the result was decided by the voters of Bradford county, or any of a hundred other subgroups. This is the case when an election is very tight.
Florida state law requires that postal ballots have a clear postmark. One of the many contentious points in the recount process was how many postal votes should be rejected for not bearing a postmark, or the postmark being unclear. Republicans argued that the application of this law was disenfranchising soldiers stationed abroad. Democrats said that it was the simple application of a pre-existing law.
This was discussed in the New York Times "How Bush Took Florida Mining the Overseas Absentee Vote"
I am not sure about an absolute decisive vote part, but Romanian presidential elections (2014) were virtually decided by Romanian diaspora:
Romania's large diaspora of up to four million people played a key role in the election. Many expat voters were said to be disillusioned with Mr Ponta.
Current President obtained almost 90% of the votes in diaspora.
BBC article mentions about protests related to not being able to vote in certain foreign countries. These protests amplified by social media acted as a catalyst for vote turnover increase. I remember that media reported that some diaspora members called their friends and relatives in Romania and insisted to go to vote against the social-democrat candidate.
So, there is a good chance that diaspora decided the actual outcome, not only directly through vote (~90% against), but also by influencing internal votes.
Ecuador is a South American country with a population of 16.4 million.
source: World Bank
There are 1.1 million Ecuadoreans living in three other countries:
It's common for Ecuadorean presidential candidates to campaign overseas.
source: CSMonitor
I don't have exact voting figures, but with such a high population in the diaspora and candidates leaving the country to win their votes, I think it's safe to say that Ecuadorean expatriates make a difference in their country's national elections.
I think that the Iraqi parliamentary election in january 2005 would count. (wikipedia link: https://en.wikipedia.org/wiki/Iraqi_parliamentary_election,_January_2005)
It was an election where a large part of voters voted in foreign territory and it would likely have quite an influence on the result, though it will be hard to prove exactly how it influenced the result.
