Greg Mankiw posted about Johnny Hallyday's expatriation and what equilibrium between countries might look like.
His analysis assumes that insurance payments are either pay-as-you-go or pay first and receive benefits later. But, the nature of social insurance is two-fold:
Regardless, Social Security insurance doesn't work because the underlying risk is fairly predictable. So governments with higher social insurance costs will lose cash flows to countries that have lower social insurance costs, ceteris paribus. So, the predicted equilibrium for two countries equal in all respects other than social insurance, is that the social insurance cost differential between the two will be no more than the transaction costs of moving from one country to the other.
It's then in the higher social insurance cost countries best interest to increase the transaction costs. Which is why the US has 26 USC Sec. 877, which imposes fines on wealthy expatriates. But, then the low social insurance countries reply by not extradicting for tax evasion, which is what Switzerland does. Given this, my guess is that we're near equilibrium already, and it's only the increasing wealth disparity in the US that causes the social insurance winners to worry.