A fixed mille-rate results in taxes going up as value goes up. This might be difficult to plan for and manage. However, many cities will simply put a lien on the property and collect when the property is sold/transferred/person dies.

However, this isn't necessarily how things need to be done. In New Zealand, the mille-rate is set to clear the budget. That way, the mille-rate changes with the city's budget, not valuations.

If the city wants to do more, they put it into the budget and say "do you want to pay for it?". Then the mille-rate is changed and everyone's taxes change.

If your house changes value, your taxes don't change! Unless your neighborhood changes its relative valuation, then it might go up or down.