According to this entry on Wikipedia, the British armed forces traditionally bury their dead where they fall. What is the origin of this practice, does it still take place and how common is it amongst other countries' armed forces?
The length of time in which you have to bury someone for practical reasons is mainly affected by temperature. You can see this by looking at different cultures, where waiting a week or more may be OK for a Russian but in Muslim culture (from the Middle East) one day is the limit. Some rain forest tribes don't even take a body back to village - they literally bury it where it falls, as they just can't wait in those conditions. Since virtually all British military campaigns have been in places hotter than Britain - some much hotter - it is clear that taking bodies home would be impracticable. On top of this, you could well have to wait to the end of the battle before you could even reach the body, by which time it might well be already in a poor state and untransportable.
If we assume that nobody brought bodies back as a matter of course, then the question becomes "When did this change?"
My parents were adults during WWII and accepted burial on location as normal. They took me to see my uncle's grave in Holland as he was killed at Arnhem. They were quite shocked when people started complaining during the Falklands conflict that bodies were not being returned. So location burial was still normal in the British forces in 1982. However, it may be significant that, even then, the Falklands were too far away to get home by plane.
On the other hand, the Americans, who had followed the traditional practice up to WWII, did return their bodies from Vietnam. I suspect this was the first major conflict where this occurred, but if anyone knows better, please post. Vietnam is about the same distance from the US as the Falklands are from the UK. I guess the logistical difference was that they had airfields in Vietnam capable of taking long-distance planes, but not in the Falklands. They were being supplied by plane and so the bodies were simply put on the returning planes. We did not have access to the only airport and so our campaign was almost entirely transported by ship.
Another important point: no one knows who you are and where your ship your body in a regular battle. Even if your corps remained in a relatively good condition, no one blows your head off or cut it in half, even if you were not covered in a huge pile of dead bodies or had your face burnt or whatever else could happen in a battle, people didn't know each other on first name basis even in the smallest conflicts. Only exceptions are maybe sieges of small places, where the defenders are relatively close acquaintances or when small raiding parties fighting somewhere.
For this very reason, people started using dog tags on their own / volunteer basis in the second half of the 19th century and it becomes a common practice at around WWI: https://en.wikipedia.org/wiki/Dog_tag
I am not an expert.
Clarence David Mackenzie (1849-1861), a drummer boy in the 13th New York Volunteers, was accidentally shot in Baltimore and his body was packed in ice and shipped back to Brooklyn.
I once read that as the Civil War went on a large number of Union dead were embalmed and shipped back home for burial. But I don't have a source for that.
The dead were buried hastily and badly near where they died on a number of battlefield in the Civil War.
The Union government established a number of national cemeteries near battlefields during the Civil War, and also near military hospitals.
The Arlington National Cemetery outside of Washington DC was used to bury thousands of Union soldiers who died in the major military hospitals around Washington. It became the major national cemetery for the United States.
The soldiers who died at the Little Bighorn were buried close to where their bodies were found, and there are markers that indicate approximately were each body was buried and the burial spot indicates approximately where each soldier died.