Preheat the oven to 325°F (163°C) and line a muffin tin with paper liners.
Combine graham cracker crumbs and melted butter in a bowl, then press this mixture into the bottom of each liner. Bake for 8-10 minutes until slightly golden, then cool the crusts completely.
Beat cream cheese and granulated sugar in a mixing bowl until smooth and creamy.
Add large eggs one at a time, beating well after each addition.
Stir in vanilla extract, fresh lime juice, and lime zest.
Divide the batter evenly into the muffin cups with the pre-baked crust.
Spoon blackberry puree onto the batter and gently swirl it to create a marbled effect.
Bake for 18-20 minutes, or until the edges are set and the centers are slightly jiggly.
Cool the cupcakes completely in the tin before refrigerating for at least 2 hours.
Top with fresh blackberries just before serving.