In Integration between The Education Edge and The Raiser's Edge, the Linking Grid manages what records are linked and synchronized in both modules. There is an option within the Linking Grid called Remove Ineligible Records.
When you edit records from the Link Records screen, you can remove records that no longer meet the integration setup criteria (as indicated in Integration > Set up Integration with The Raiser's Edge/Education Edge). To do this you will click the Remove Ineligible Records button. This will not remove any matched or linked records, only records that are both unmatched and unlinked.