Complexity of Block edit distance with Swapping only
When one operation is exactly "removing a block and inserting it between two other positions", the problem of computing the string distance is known as Transposition Distance. It is NP-hard even if both input sequences are $p$-sequences, that is, if every letter occurs only once in each input sequence:

Laurent Bulteau, Guillaume Fertin, Irena Rusu:
Sorting by Transpositions Is Difficult. SIAM J. Discrete Math. 26(3): 1148-1180 (2012)