I usually wash the greasy lunch box under warm water with the detergent on a sponge. Clean the greasy box under warm to slightly hot water to get rid of most of the residue and excessive oil/fat. Then use the sponge to wash thoughtout the box in particular the bottom. You have to create foam during this process since it is the foaming effect that emulsifies the grease. Wash under running warm water and dry. If it is too greasy, I will leave the detergant there for a couple minutes before washing.
To wash lunch box after storing fatty food requires some work, so if you would please something like a piece of flat leave lettuce or a layer of rice to separate the food from contacting the bottom, it will ease your work. A plastic wrap also works but I rather choose to use lettuce or rice which are more natural.