N.H. Rev. Stat. Ann. § 289:17

The monuments, gravestones, and other appurtenances attached to the graves shall be carefully removed and properly set up at the place of reinterment with as little injury as the nature of the case will admit. In case of injury to any monument, gravestone, or appurtenance, the damages shall be assessed by the governing body in the same manner and with the same right of appeal as in the case of alteration of the grade of highways.