Noah
Informing the DWP of a change of address should not have any effect on your ESA or PIP, however, if you move Local Authorities and make a new claim for Housing Benefit then this may trigger a claim for Universal Credit if the new area is a UC full Service area, any Income Related ESA that you would receive would then be moved to UC as well. This should not require a reassessment of your claim.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems 
