Hi Robin
Thank you for your question, unfortunately the answer is not straight forward.
If you require help with rent, if you move from one local authority to another local authority and you are not in supported accommodation then you will have to apply for UC to claim the Housing Element.
If you do not have to claim help with rent either private or social landlord then you can still stay on legacy benefits.
Gary
Nothing on this board constitutes legal advice - always consult a professional about specific problems