Bring the blueberries and cherries to a boil in a large saucepan until they pop or release their juice. Hull the strawberries and cut very large ones into quarters. Add the more delicate fruit like strawberries and raspberries last. Stir in the syrup and sugar to taste.
Dissolve the cornstarch in at least 1/4 cup cold water. Remove the pan from the heat, stir the cornstarch into the fruit mixture, and cook briefly over low to medium heat, stirring constantly. Make sure not to undercok the pudding, otherwise it will taste chalky. When the pudding turns clear and thickens, remove the pan from the heat immediately. Continue stirring for another 1 to 2 minutes.
Pour the hot pudding into a glass serving bowl or individual dessert bowls. To prevent the glass from cracking when you pour the hot pudding into it, put a damp dishtowel underneath the bowl. Refrigerate for several hours until set. Serve the pudding cold, but take it out of the refrigerator 30 minutes before serving, so it can develop its full flavor.