If a person travels from one country where he started fasting at the beginning of the month, to another country where Eid al-Fitr comes later, then he should continue fasting and should not break the fast until they do. 

Shaykh Ibn Baaz (may Allaah have mercy on him) was asked: 

I come from eastern Asia, where the hijri months start a day later than in the Kingdom of Saudi Arabia. I am going to travel to my homeland during Ramadaan. I started fasting in Saudi, but at the end of the month we will have fasted thirty-one days. What is the ruling on our fast? How many days should we fast? 

He replied: 

If you start fasting in Saudi or anywhere else, then you fast the rest of the month in your homeland, break the fast when they do, even if that is more than thirty days, because the Prophet (peace and blessings of Allaah be upon him) said: “The fast starts on the day you fast and the breaking of the fast comes on the day you break the fast.” But if you have not completed twenty-nine days of the month, then you have to complete it, because the month cannot be less than twenty-nine days. 

End quote from Majmoo’ Fataawa Ibn Baaz, 15/155 

Shaykh Muhammad al-Saalih al-‘Uthaymeen (may Allaah have mercy on him) was asked: 

What is the ruling on one who starts fasting in a Muslim country then goes to another country where the people started fasting later than the people in the first country, when following them means that he will fast for more than thirty days or vice versa? 

He replied: 

If a person travels from one Muslim country to another, and the breaking of the fast comes later in the country to which he has traveled, then he should continue to fast with them until they break the fast, because the fast is the day on which the people fast, and the breaking of the fast is the day on which the people break the fast, and the sacrifice (al-adha) is the day on which the people offer the sacrifice. He should do this even if he fasts an extra day or more. This is similar to the case if he travels to a country where the sunset comes later; he should continue to fast until it sets, even if that adds two hours to his day, or three, or more. This also applies if he travels to another country where the new moon has not been sighted, because the Prophet (peace and blessings of Allaah be upon him) commanded us not to fast or to break the fast until we see it. He said: “Fast when you see it, and break your fast when you see it.”

With regard to the opposite case, which is when a person travels from a land where the onset of the month comes later to a land where the onset of the month comes earlier, then he should break the fast with them, and make up any days he has missed of Ramadaan later on. If he misses one day, he must make up one day, if he misses two days he must make up two days. If he breaks the fast after twenty-eight days, then he must make up two days if the month was the full thirty days in both countries, and he must make up one day if it was only twenty-nine days in both countries or in one of them. 

Majmoo’ Fataawa al-Shaykh Ibn ‘Uthaymeen, 19/ question no. 24. 

He was also asked: 

People may say: Why do you say that he has to fast more than thirty days in the first case and make up missed days in the second? 

He replied: 

He has to make up missed fasts in the second case because the month cannot be less than twenty-nine days, and he should fast more than thirty days because the new moon has not been sighted. In the first case we say to him: Break the fast even if you have not completed twenty-nine days, because the new moon has been sighted, and when the new moon has been sighted it is essential to break the fast, and you cannot fast on the first day of Shawwaal.  If you have fasted less then twenty-nine days, you have to complete twenty-nine days. This is unlike the second case, where it is still Ramadaan for you when you come to a country where the new moon has not yet been sighted. It is still Ramadaan for you, so how can you break the fast? You have to continue fasting. If the month becomes longer for you, then it is like extra hours in the day. 

Majmoo’ Fataawa al-Shaykh Ibn ‘Uthaymeen, 19/ question no. 25 

See also the answer to question no. 38101. 

And Allaah knows best.