Shortest number of editing move between two words
I don't know whether this exact problem has been studied, but  Chaudhuri et al. studied the related tandem duplication-random loss problem: you are given a permutation, and you want to transform it into the identity permutation by 
 (1)  duplicating a segment of any length and appending the copy right after the original, then (2) deleting elements so that you obtain a new permutation instead of a string. Note that applying (1) then (2) accounts for one operation. 

Different variants can be defined according to the weight given to each operation, which in their paper depends on the width of the duplicated segments. 
They also study a similar problem with the whole genome duplication, which is exactly the kind of duplication you allow. I don't remember reading about work on this problem in the context of strings, but I hope this can at least give you a starting point for your searches.