In a large saucepan, bring cranberry juice, cider, and sugar to a boil over medium-high heat. Cook, stirring, until sugar is dissolved, 2 to 3 minutes. Remove from heat; stir in lemon juice.
Transfer to a 9-by-13-inch baking dish. Freeze for 2 hours, then mash with a fork. Cover, and continue freezing until set, at least 2 hours and up to overnight.
Transfer sorbet to a food processor; puree until smooth. Transfer to an airtight container; refreeze until ready to serve.
Apple juice may be substituted for apple cider, although cider will give you richer flavor.