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I'm sure it depends on the jurisdiction, but all the property taxes I have ever paid (which, to be fair, has been in one county) have been a fixed percentage of the value of the property.

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Those percentages tend to change over time though, right? What inputs do you think they have to their ideas on how to adjust it?

I don't think the percentage has changed since I moved here, though the dollar amount I pay has gone up significantly as the assessed value of my house has risen.

Like with any tax, I'm guessing the rate would change if the county's revenues didn't match their expenses?

In my county, tax rates have to be renewed every year. So every year there's a chance for the rates to change. There is an automatic suggestion of a "no new revenue" option formulated by the appraisal district based on their new assessments, there are then voter approved changes, and then the county (or taxing entity) decides.

My property taxes have practically only gone up over the years, but the tax rates from the various entities have mostly trended downwards.

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