Heat the oven to 350 degrees F and line a baking sheet with parchment paper. Mix the flour, brown sugar, cinnamon, sea salt, and dairy-free butter until large crumbs form.
Bake the cinnamon crumble for 11 to 13 minutes, stirring once, until set and lightly browned. Cool completely, then place the crumble in the freezer.
Combine the brown sugar, dairy-free butter, oat milk, cinnamon, and sea salt for the swirl in a small saucepan. Simmer for 2 minutes while whisking, then transfer to a bowl and refrigerate until thick and cold.
Whisk the coconut milk, oat milk, cane sugar, cornstarch, coconut oil, vanilla, and sea salt in a saucepan. Cook over medium heat until the mixture is steaming and slightly thickened, then refrigerate for at least 4 hours.
Churn the cold ice cream base according to the ice cream maker's directions. Add two-thirds of the frozen cinnamon crumble during the final 2 minutes.
Spoon one-third of the ice cream into a freezer-safe container and drizzle with cinnamon swirl. Repeat the layers twice without stirring them together.
Scatter the remaining cinnamon crumble on top and freeze for 4 to 6 hours. Soften the ice cream at room temperature for 5 to 10 minutes before serving.