Had Flavius become the new Caesar? If so, was it due to it's usage as a praenomen (or nomen? did it change since the Flavian dynasty?) by the Constantinian dynasty?

The first post-Constantinian Emperor, Jovinus, has Flavius and Augustus as parts of his name, but is the first (I think) to not use Caesar- is this evidence for the shift in usage?

Also interesting to note that Constantine III, son of Heraclius, and first Emperor to not have his reign officially recorded in Latin (since Heraclius changed the language of government to Greek in 620), doesn't have Flavius as part of his name.

My source for all of this is wikipedia