Council Tax Support is usually linked to your address, so it may stop when you move and you will normally need to apply to your new local council. Tell both councils your moving date, including your old and new addresses, and check whether you need to claim again. Your new council will assess your application under its own scheme, based on your income, household and circumstances. Apply promptly, as backdating is not automatic; check the relevant council’s website or GOV.UK for its rules.