Put cherries in a small saucepan. Add 1/4 cup water and bring to a boil over high heat. Lower heat to medium and cook, stirring, until cherries have released their juice and pulp has softened.
Transfer cherry mixture to a blender and pulse 3 or 4 times to purée fruit pieces. Pour through a strainer into a medium heatproof bowl, using the back of a ladle to push any juice and purée through. Discard solids. Add sugar to hot liquid and stir until sugar has dissolved completely. Cover and chill for at least 1 hour.
Add milk, kirsch, and lemon juice and stir to combine. Pour into an ice cream maker and churn according to manufacturer's directions. Transfer to an airtight, freezer-safe container and freeze at least 4 hours.
Note: Nutritional analysis is per 1/2-cup serving.