We don't have a property tax in sweden.

Also no inheritance or gift tax. Very good place to live if you are wealthy and don't make money the peasant way of going to work.

I see your problem right there...

Why did a lot of rich people (Bjorn Borg, Ingvar Kamprad (IKEA)) move out of Sweden, then?

They moved out earlier, when there were (quite high) property taxes. The tax was abolished in 2007.

Ingvar Kamprad moved to Switzerland in 1976; in 2014, when the abolition of 2007 had survived one election cycle, he moved back to Småland.

Björn Borg moved out of Sweden already in 1974 at age of 18, perhaps more because on 90% income tax than property taxes.

This totally depends on how your municipality levies property taxes. In many parts of the Northeast US, for example, the county first sets its budget for the year, and then levies tax by dividing the budged among property owners in proportion to assessed property value. Therefore, if the county's budget hasn't changed, and if all of your neighbors' houses have increased in value by the same % as yours, then you'll pay the same tax as last year. Conversely, it's possible for your house to increase in value but to owe less tax if your neighbors' houses increased in value by more than yours.

This is very location-dependent. Here in the UK, the nearest thing to a property tax is the "council tax", which is based on the estimated value of the property in 1991.

There is also a tax on property transactions which can be considerable.

Yeah. This 100%. It's a broken system for everyone except investors (and maybe even for them, ultimately).