No, they are not constant, and go up and down with rates. However, they are based on a factor of the purchase price, not the current market, like rent or a new purchase.

However, rates do go down often. We're at 15 year highs now, so the highest for most mortgages, and probably the most any have paid. However they are expected to drop next year again. Rents won't. Those paying the highest they have ever paid 10+ years into their loan are still paying less than the equivalent rent they would be paying if they didn't own.