Break grapefruit and tangerines into sections. Cut grapefruit sections into thirds; cut tangerine sections in half. Transfer grapefruit, tangerines, and all juices to deep serving bowl. Using small sharp knife, cut all peel and white pith from oranges. Slice oranges into 1/4-inch-thick rounds, then cut slices into quarters. Add oranges and all juices to same bowl. Mix in 1/2 cup dried cranberries, honey, and cinnamon. Cover and refrigerate at least 1 hour. Mix yogurt and ginger in bowl. (Fruit and yogurt can be prepared 1 day ahead. Cover separately; chill.)
Spoon yogurt atop fruit. Sprinkle with brown sugar and dried cranberries.