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"