In a medium saucepan, over medium heat, cook the berries until they are soft, 15-20 minutes. Remove from heat, put softened berry mixture into a blender, and puree. Let cool to room temperature.
In a large glass measuring cup, stir together the yogurt, honey, and lemon juice. Swirl the cooled fruit mixture into the yogurt mixture, creating a marble effect.
Pour the yogurt-berry mixture evenly into 10 ice pop molds. Cover molds with lids and insert wooden craft sticks, about three quarters the way down.
Freeze until firm, about 4 hours or overnight. When they are fully frozen, run the bottom of the4 molds under warm water to easily remove the ice pops. Place them in a baking pan or half sheet pan lined with parchment paper and put immediately back in the freezer to harden back up.
Serve immediately, or store the pops in the freezer, in freezer-proof bags for up to 1-2 months.