But I couldn't think of very many places that originated their own team sports. So I went googling about. Yikes: the things I didn't know were legion.
Some we all know: In North America/Mexico they had lacrosse and its variants ("each team consisted of about 100 to 1,000 men on a field that stretched from about 500 meters to 3 kilometers long") and ulama. In Persia a cavalry training exercise seems to have developed into polo.
Now I got into new territory. I gather that the Greeks and Romans had something I'd not heard of before, episkyros/harpastum; though I don't see any signs that these survived past the collapse of Western Rome.
But baseball/rounders, cricket, soccer, rugby, American football, Australian football, basketball, hurling, modern volleyball, modern hockey (though there are very ancient precedents) and so on seem to come from the sports-mad British Isles and their descendents. Even more new team sports were invented last century (Switzerland, Sweden, and so on).
(I'm not going to include the battledore, because it seems not to be exactly a team effort, though it was widely popular from antiquity. South Africa has a booklet of traditional games, including board games, stick fighting, etc: but I can't tell if the dibeke is old or new.)
A few of these are pretty explicitly connected to warfare (polo), and old lacrosse and ulama seem to have had religious meanings, but most of the rest seem to have lost those connections, if they ever had any. Given simple high spirits and a few rules to give shape to the interaction (no Calvinball here); if the rules worked out well enough you'd have a traditional game.
Some sections of the globe are missing from the list. Everybody plays team sports now, but why didn't some cultures seem to develop any themselves? Or maybe they did develop some which fell out of favor with changing culture, or maybe the climate changed and they didn't have as much leisure (I'm thinking Amazon here--it looks like they used to build large farms there once upon a time). Or maybe the culture associated sports with childhood, and the visitors who documented the culture didn't pay attention to the kids.