A change of address does not affect your ESA. However if you are claiming Housing Benefit you may only be able to transfer the HB claim if your new address is in the same local authority. If you are moving to a new local authority, in most cases, you will not be able to make a new HB claim. If that is the case and you want help with rent you would need to claim Universal Credit.The act of claiming UC would then end your income based ESA. If you have contribution based ESA entitlement that would continue alongside the UC.