Make sorbet syrup: In a small saucepan over medium heat, cook sugar with 1/3 cup water, swirling pan, until sugar dissolves, 3 1/2 minutes. Cool completely.
Make raspberry layer: In a food processor or blender, purée raspberries with 1 1/2 tbsp. water and the lemon juice. Strain into a liquid measuring cup, then stir in 3 tbsp. sorbet syrup. Using a small funnel, fill "nose cone" of each popsicle mold with 1 to 2 tbsp. raspberry mixture, depending on size of mold. Freeze, nose-cone side down, until firm, about 40 minutes. Rinse food processor, strainer, measuring cup, and funnel.
Make coconut layer: In food processor, blend coconut milk and lime juice. Pour into measuring cup, then stir in 2 tbsp. sorbet syrup. Using funnel, pour 1 to 2 tbsp. coconut mixture to second "fill line" in mold, on top of raspberry layer. Freeze until firm, about 40 minutes. Rinse food processor, cup, and funnel.
Make blueberry-lime layer: Purée blueberries with lime juice. Strain into measuring cup, then stir in 1/4 cup sorbet syrup. Using funnel, pour enough blackberry-lime mixture into each mold, on top of coconut layer, to come about 1/4 in. below top. Insert sticks into each popsicle, pushing them into coconut layer. Freeze until solid, at least 3 hours.
To eat, run molds under water for a second, then gently pull popsicles free.
Make ahead: Sorbet syrup, up to 1 month, chilled airtight. Pops, up to 2 months, in molds.
Nutritional analysis is per 2 1/4-oz. popsicle.