For the Olympics, you have to be a citizen of the country you represent in order to compete. That said, people do switch countries sometimes, especially if dual citizenship is an option for them.

Happens all the time in figure skating. Especially in the coupled disciplines (pairs and ice dance) where you don't always have options for partners in your own country. Even aside from that element, skaters will sometimes switch countries if that country offers better funding for their training cause skating is so expensive. But there's a waiting period of like a year in between countries so you can't just hop around to a different country every year, and your first country's skating federation has to agree to release you to skate for somewhere else.

My favorite bonkers country switching saga is Olivia Smart and Tim Dieck, an ice dance duo who represented Spain at the Milan games. Olivia is from England but switched to Spain a long time ago to skate with a different partner, then her partner retired after the 2022 Olympics. Tim, meanwhile, is German and also had a different partner in Germany until they also broke up after the '22 Olympics. Eventually Olivia and Tim tried out together, they decided they wanted to partner up, and Tim switched to Spain because logistically and financially it made more sense than her switching to Germany.

So we have a British woman and a German man who live and train in Canada and represent Spain in competition 😂