Basically exactly what the question says on the tin.
Were they always considered Chinese? Was the history repurposed to make China look stronger? Is it something as simple as they won the mandate of heaven so are therefore Chinese?
Also very curious as to when the shift happened if they started out as foreign powers and later came to be regarded as Chinese, especially if that happened during their reigns
Well, they have been viewed as both, sometimes – seemingly contradictorily – at the same time. I discuss dimensions of this in a few prior answers:
Though to be fair, I have yet to write a complete answer on the topic. If you have any specific follow-ups in order to clarify that, I'd be more than happy to reply to them.