I recently got a new phone. When getting rid of my old phone, I transferred all of my world files via USB to my computer from the following folders:
Internal storage > games > com.mojang > minecraftWorlds
In my new phone, when I went to put the worlds in, I couldn't find the games folder. When I searched, I couldn't find com.mojang The closest result was Android > obb > com.mojang.minecraftpe. This was an empty folder.
My old phone had Android Lollipop (Moto e2). My new phone is Android Marshmallow (ZTE Prestige 2).