As a selfemployed I do not get any pension at all. That's the point. I'd be in for a tax based pension system and I'd happily pay for it. Unfortunately public pension is prohibitively expensive and a pretty bad investment. You're better off investing your savings yourself. So there's no point in waiting for me.

Most countries still pay out partial pensions if you live there for a number of years and reach their retirement age after moving to some other place.

Really? Which are those countries? If you never pay into public pension, how can you get a pension then?

If you never pay into the system, then obviously you don't get anything from it. But if you do pay in, then my understanding is that they'll send you your pension money to wherever you live. That's how it works in the EU, UK. Depending on the country, there can be minimal number of years of work required for the pension to trigger, but AFAIK in the EU they're counted across the whole Union and not per-country.

BTW this is not a verified information so please verify all of the above yourself if you intend to depend on it in any capacity.