If their only qualification is that of being named on the register of electors, they must resign if they cease to be named on the register. If however, they satisfy any of the other qualifications (i.e. occupying as owner or tenant land or buildings in the town during the whole of the 12 months before the day of nomination and before polling day; main or only place of work in the town during the 12 months prior to nomination and prior to polling day; or living in the town, or within three miles of it, during the whole of the 12 months before the day of nomination and before polling day) they are not required to resign if they move away (assuming they included these additional qualifying criteria on their original nomination form.)
This rule has been a source of great frustration to many, especially in a situation where members of a Council are entitled to claim travel expenses for journeys from their home address to a meeting of the Council. I believe that the record for a County Councillor stands at a round trip of almost 8,000 miles! Let us hope that common sense will one day prevail and local government can mean just that.