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.