Over a large bowl or pan, add both ingredients to a food mill. Run the applesauce and berries through the food mill, until only the blackberry seeds remain.
If you don't mind seeds in your fruit leather, simply add both ingredients to the bowl of a stand mixer, and beat on medium for about 3 minutes until blackberries are well blended into the sauce.
Spread the mixed puree on the fruit leather trays for your dehydrator. (This recipe nicely fills three trays on my Nesco FD-61). For a good result, it's important to spread the puree evenly, with no thin or thick spots. Spreading it perfectly flat and even is the trick to perfect fruit leather!
Dehydrate at 135 degrees for about 6-7 hours. Fruit leather will be done when it's fairly dry to the touch and feels like soft leather. It will be a bit glossy, and shouldn't leave any indent when you touch it.
Allow to cool, then peel off dehydrator trays. You can easily tear it by hand, or for tidier strips, use scissors or kitchen shears to cut to the desired size.
For OVEN drying, spread puree evenly onto parchment-lined baking sheets. Dry at 140 or the lowest temperature your oven allows. Timing will be less consistent than in a dehydrator, so start checking on it after a few hours, then keep checking every half hour til it's done. You'll be looking for the same signs of "done-ness" as with the dehydrator method.
Store in air-tight container at room temperature for up to 1 month, or store tightly-sealed in the freezer for longer storage.