MsJane
Moving house will not cause your ESA to be migrated to UC, however, if you are receiving Housing Benefit then his may cause a transfer.
If you are receiving HB and move within the same Local Authority then you will not be required to claim UC.
If you move to a new LA and your ESA or HB includes the SD Premium then you are currently barred from claiming UC and will be able to make a new claim for HB but be aware this restriction is due to end in mid January.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems