Not sure for the others, considering the only that real have a claim for me is the Frankish Empire, the Byzantine Empire obviously and HRE, the others are not really successors of any kind.What are the arguments for the HRE? Or the others?
The Holy Roman Empire claim is that they share the same religion(Nicean Christianity in this case), ruled North/all Italy, used latin, continued Roman-created institutions etc.
And for the HRE those reasons start to weaken in the 15th century and disappear by the end of 16th century.