Special equipment: a dehydrator with a non-perforated insert
Combine the raspberries, strawberries and 1/2 cup applesauce in a blender and blend until very smooth. Strain through a fine mesh sieve and transfer to a medium bowl or measuring cup.
Combine the pineapple and remaining 1/2 cup applesauce in the blender and blend until very smooth.
Put one nonperforated dehydrator insert on a work surface. Pour the pineapple puree onto the insert in random dots. Pour the berry puree onto the insert to fill the remaining space, making sure not to overlap too much with the pineapple (it will create thicker layers that won't dry properly). Gently shake the insert to evenly spread the puree all the way to the sides. Use a spoon or offset spatula to thin out any thick spots.
Place the insert in a dehydrator and dry according to the manufacturer's instructions at 125 degrees F until the fruit puree is dry to the touch, matte and no longer shiny, about 12 hours (times may vary depending on the machine). Test with your fingers to make sure it is dry all over. Let cool at room temperature for 5 minutes. Use a butter knife to loosen the fruit leather from the insert and slowly peel away.
Cut or tear the leather into pieces and store in an airtight container up to a week.
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