Place avocados, lemon juice, garlic, salt, and pepper in a resealable plastic bag. Seal bag and knead bag to mash mixture.
In a 6-cup straight-sided clear glass bowl, spread the black beans in an even layer. Spread avocado mixture over black beans. Spoon white beans in mounds over the avocado layer; carefully spread evenly. Spoon salsa over white bean layer; spread evenly. Spoon mounds of sour cream over salsa layer; carefully spread evenly. Sprinkle with cheese. Cover and chill up to 24 hours.
Spoon layered dip onto tortillas, roll up, and serve.