Hi davis
If you move within the same local authority and have to claim help with your housing costs then you will stay on legacy benefits. If you are told otherwise the you can appeal the decision, at that stage I would advise you to seek help from your local Welfare Rights Organisation;
advicelocal.uk If you move to a different LA and need to claim help with your housing costs then that will be a change of circumstances and you will have to apply for UC to get the housing element.
If you move to a different local authority and you do not need to claim help with your housing costs you can still stay on IR ESA, if you are on IR ESA.
Gary