Using on/off turns, coarsely chop cranberries in processor.
Transfer cranberries to medium bowl. Mince half of kumquats, removing any seeds. Thinly slice remaining kumquats, removing any seeds.
Mix all kumquats, sugar, and ginger into chopped cranberries (as mixture sits, sugar will dissolve). Cover relish and refrigerate at least 4 hours. (Cranberry relish can be prepared 1 week ahead. Cover tightly and keep refrigerated.)
