Here is the current (2019) list of the ten largest countries by GDP (purchasing power parity), courtesy of Wikipedia:
1. China
2. USA
3. India
4. Japan
5. Germany
6. Russia
7. Indonesia
8. Brazil
9. United Kingdom
10. France
The last two are by far the hardest and probably require a POD before 1000. The other eight are all doable.
CHINA
Beijing and precursor cities were the Mongol and Jurchen capitols, and useful for sanitized nomad dynasties ruling China because it was just within historically settled China but also relatively close to the center of gravity of the nomad empires. In the early 15th century the Ming, after using Nanjing as the capitol, selected Beijing because they wanted to prioritize a repeat of the nomad conquests. Because the Qing (Manchus) were the ultimate sinicized nomad conquerors they kept the capitol in Beijing, and then path dependence has kept it there since. Beijing made history as the first Chinese capitol not to be burned to the ground when its dynasty ended.
Previous Chinese capitols had been more in central China, generally moving from west to east, Xian, Loyang, Kaifing, Linan and Nanjing. The Ming were the first native Han dynasty to unify China and select Beijing.
If you make the Song militarily stronger and have them defeat the Mongols and the Jurchen, or remove Ghengis Khan, then very likely the Song capitol of Kaifing remains the capitol. When the Song ends they are just replaced by a native Chinese dynasty. The post 1900 alternative is to change Chinese history so that Beijing gets destroyed so thoroughly that instead of rebuilding it, the new government that unifies China decides to make the most obvious alternative, Nanjing, the capitol. Or add extra crazy to the cultural revolution and they go with a planned capitol built from scratch, though sometime after Mao's death this probably would have been abandoned.
USA
While there had been plans kicking around for a planned capitol, the impetus to adopt them was a scheme to put federal finances in order by stiffing people who had lent the government money during the war of independence. Since the wealthiest people in the USA at the time were slaveholders and they were hurt financially by the scheme, and the South at the time provided most federal revenue, it was decided at the time to just build the planned capitol in the South, but as far north as possible. The South had no real cities other than Charleston, which was too far south and still smaller than New York, Boston, or Philadelphia. Path dependence then keeps the capitol in Washington.
Find some way to avoid the maneuvering I just described and the capitol stays in Philadelphia.
Once its in Washington I don't see any way of getting it away from there. It becomes a matter of national pride to return the capitol there and rebuild the city if it's destroyed. The British in fact did destroy it in 1814 and the Americans returned and rebuilt. There was no real chance of rebel armies capturing it in the Civil War, but given that Congress expanded the Capitol Building during the war you would have just had the city recaptured and rebuilt anyway. Same with nuking it during the twentieth century. And if you were going to put your capitol in a slave state but close to the North, which was the whole point, there was really no better site.
INDIA
Delhi, like Beijing, is in a good position to keep an eye on the most likely invasion route into India, or if your dynasty was one of the invaders, to keep in contact with your homeland. Its tended to be the capitol of the largest Indian state since at least the 11th century, before it was in Bihar. The British built a new administrative center north of Delhi, New Delhi (though its really just one big city) and the post-Raj government moved into the offices.
Technically you could have the British not build New Delhi and the capitol remains in Delhi. Other than that, your best bet is to somehow make the area Muslim so the entire Punjab and Delhi itself goes to Pakistan after the partition. Or have the "Sultans of Delhi" base themselves in a nearby city, with Lahore or Agra the likeliest possibility. The Sultans tried to move the capitol to Deccan several times but all these attempts resulted in disaster. The best way to do it would be to have Lahore be the main capitol and center of the Raj, and when it goes to Pakistan the government of (Hindu) India builds a planned capitol or selects another city. You could also have the British continue administering the Raj from Kolkata, which is probably the default capitol of India if you remove Delhi somehow.
JAPAN
Obviously Kyoto, or Osaka if the rulers want a more modern city, and you do it by preventing the Edo (Tokyo) Shogunate. If the Shoguns are based in Osaka then that will become the capitol of Japan. If you want a planned capitol in an small village, the late twentieth century Japanese government had schemes to move ministries out of Tokyo to stimulate regional economies and reduce overcrowding in Tokyo, so just have them actually go ahead and do this and build an entirely new capitol for the economic stimulus.
GERMANY
Berlin was the capitol of the smaller kingdom that unified Germany. The Hohenzollerns could have picked another city in Brandenburg, such as Magdeburg. Other alternatives are German unification driven by some other kingdom, the 1848 Frankfurt parliament succeeds, either the 1919 or 1949 republics adopt Frankfurt, or they could even decide in 1991 that they would rather stay in Bonn.
RUSSIA
Moscow was the capitol of the smaller kingdom that created the Russian empire. Its sort of the same situation with Berlin. Have the Muscovite Tzars be based somewhere else in the areas such as Vladimir or Smolensk, or have some other dynasty found the Russian empire. They moved the capitol to Petrograd for two centuries so it could also just stay there.
INDONESIA
I know very little about Indonesian history, but Jakarta/ Batavia just happened to be the place that the Dutch East India Company had as its main base, so any other city in Java, or Palembang on Sumatra, could have fulfilled that role. And we are still assuming that the Dutch and not some other colonial European power creates modern Indonesia.
BRAZIL
The Portuguese originally governed Brazil from Salvador in Bahia, and moved it to Rio de Janeiro when it was clear that the southern provinces had more economic potential. There had long been an idea of a planned capitol in the geographic center of the country (though far from the population centers) and it was written into the 1889 constitution. The winner of the 1959 presidential election promised to actually build the city, for various reasons, and he did so, and we got Brasilia.
Its not difficult to keep Kubitshek from power and keep the capitol in Rio de Janeiro. If the center of Rio is though too overcrowded, just move the government offices to the Zone Ouest/ Barra de Tijuca area which the government was anxious to develop anyway. They could even build the same buildings that were built in Brasilia there. The Braganzas also came very close to selecting Sao Paulo as the capitol so that is another obvious alternative to Brasilia.
UNITED KINGDOM
As I've commented elsewhere, this is really hard because London is where you cross the Thames when arriving from or leaving the continent, and the part of the continent where the English did most of their trade, by the most direct route. It was the capitol and largest city in England in Roman times. And no, preventing the Norman conquest doesn't work since the Saxons were already building what became the Palace of Westminster and Westminster abbey. And preventing the rise of Wessex doesn't work since London was part of Mercia when Mercia was dominant. The only way I can see this working is for the Norse and/ or Danes to concentrate their expansionist energies on England for some reason and turn the place into an extension of Scandinavia, in which case the link to the continent becomes less important and you probably see the capitol in York or Lincoln. While this is much less likely, you can have Charles I win the civil war, but have to besiege London, so after he wins he decides to just keep the capitol in Oxford.
FRANCE
This is also difficult, but not as difficult as with England. The Dukes of France created the Kingdom of France and they were based in Paris. However, Soissons was an alternative, though not as good as Paris, or just have some other dynasty than the Capets, based elsewhere, create the Kingdom of France.