There are a number of tax relief programs available to seniors that, under certain circumstances, can prevent a property tax increase if they sell their existing home and buy another one.
Proposition 193 allows the new property owners to avoid property tax increases when acquiring property from their grandparents.
Links
[1] https://sfassessor.org/../../tax-savings/tax-relief/transfer-assessment-replacement-property-senior-citizens
[2] https://sfassessor.org/../../tax-savings/exclusions/reappraisal-exclusion-between-parent-child
[3] https://sfassessor.org/../../tax-savings/exclusions/reappraisal-exclusion-grandparent-grandchild