1) A citizen of a country commits a heinous crime. Instead of sending the criminal to jail to be supported or even executed on the taxpayer dime, the country simply strips the criminal of citizenship, and deports them at the nearest border.
2) A citizen of another country has a track record of exceptional and quantifiable contributions. The "sports team" country decides to "recruit" this person and people like this person, offering them generous citizenship packages, in exchange for ensuring future contributions are made to the "sports team" country.
One catch I can see is that "sports team" country has to have the right metrics to determine who gets recruited and who gets booted. It would have to be some sort of quantifiable metric that shows how "profitable" each citizen is to the country, like:
Tax Income Received - Cost of Government Services Used
Without quantifiable metrics, the "sports team" country at best would be hiring and firing on anecdotes, and at worst, would be hiring and firing for silly political or bureaucratic reasons.
At the end of the day, I think it'd take trial and error to get all the details of a "sports team" country right. But this mentality seems better than stocking your bench with an at-best unproven line-up, all in the name of sympathy.