Judy
It's a misunderstanding that a CofC results in a move to UC, however, some CofCs can necessitate a claim for UC at the moment the only one is being in receipt of Housing Benefit and moving to a new Local Authority.
Any existing HB claim cannot be moved to a new LA (note:: moving within your existing LA is covered as a change to HB), and as you can no longer make a new claim for HB the only way of covering housing costsin the new LA is via the Housing Element of UC.
Changes such as a change in you health conditions will result in an assessment but under your existing ESA claim.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems 
