Why should property tax change when value changes? Because it's a tax on capital. The basic tax is a percentage of assessed value. Prop.13 works several ways to avoid abrupt changes to a property owner's taxes: it limits the change in assessed value to a 2% increase per year, regardless of how much the value actually goes up; it limits the base tax rate to 1% of assessed value, plus other things you voted in, such as bonds that need repayment, and non-value-based things like Mello-Roos and special district charges; and it requires a 2/3 vote (property owners or legislature) to increase or apply a new tax. But it's still a percentage of property value, so when the value goes up so does the tax bill. And if you do something that affects value, like add on or build a swimming pool, expect the value of that (you got a building permit, right?) gets added to the assessment right away. Overall, it keeps the changes in property tax small and predictable as long as you own the place.

The main "out" in Prop.13 is change of ownership. Sell the place, and the assessment for the new owner is what is paid for it. So the people who bought into a neighborhood recently pay a dramatically higher tax than those who have lived there a long time. Yes, that's arguably unfair because those who have lived there a long time still use services that need to be paid for. Places like San Francisco with high turnover and high (and rapidly growing) prices are heaven for California property tax collectors; places with little turnover and low prices (many rural communities) can't keep up with inflation and see declines in the ability of local government to provide services without other taxes or outside help.