Taxes track a base year, this explainer not today’s price. A base year sets the number and a sale resets it.
That is why two identical houses on one street pay very different bills. One was bought decades ago, nothing more.
After a sale a supplemental bill arrives, usually with a delay. It bills the increase for the partial year.
The family transfer rules were rewritten. Read it before making moves, as what worked before may not now.