I can tell you about the Carolingian organization and the birth of feudalism

Carolingian administration was in some aspects a direct inheritor of Roman tradition, attemtping a centralized imperial administration, where important regional powers were actually functionaries of the empire, so their authority was derived of their posting by the emperor. In this way, you had regional administrators, which were submitted to the vigilance of comes, a title of Roman origin. These comes were in charge of traveling the empire through the seasons, visiting landlords who were far away from their own jurisdictions, in order to verify that they were properly fulfilling their functions. In this sense, the person charged with performing a task, received honorum, which was the appointment to office, and beneficium, which was the benefit given together with honorum as payment. In principle, honorum was easily revokable, and was not to be inherited by the holder's offspring, but in reality, the office was almost never revoked, and was most often passed directly from father to son. This tendency was intensified as the functionaries gained more and more gravitas, to a point when revocation became not only something that the emperor didn't want to do (in most cases), but also something he couldn't do, even if he wanted. This privatization of charges came with a confusion of the concepts of honorum and beneficium, becoming one and the same, and hence feudalism was born.

Regarding the economic administration, the Carolingian Empire kickstarted the confusion of Res Publica and private business that characterized medieval politics (to some extent) and particularly, medieval treasury and economic policy.

In the matter of warfare, Carolingian forces were highly centralized, which made the military much less flexible, and given that flexibility was needed to confront the menace posed by the non-Latin Christian peoples (vikings, muslim raiders and Slavic invaders), this centralized scheme of military structure was replaced by a highly decentralized one, where regional authorities took direct charge of local forces to react quickly to threats. This capacity of the local powers to better protect their people, further helped them to consolidate and legitimate their authority of autonomous origin.

My sources are

Donado Vara, Julián, La Edad Media. Siglos V-XII

Rosamond McKitterick, Charlemagne: The Formation of a European Identity

and others that I don't have at hand to quote.

EDIT: to correct some typos.

This question is covered in some ways by Ian Wood's The Merovingian Kingdoms 450-751 and was a pretty good read. He does a pretty good job showing that there is some continuity at least initially between Rome and Merovingian Gaul. In many instances the "bureaucracy" so to speak continued to function immediately after the Western Empire dissolves, but then evolves into its own organism that is based much more centrally around monasteries (see Matthew Inne's State and Society in the Early Middle Ages). In a Carolingian context they drew on older Roman and Frankish traditions, not to mention Christian ones, in order to maintain legitimacy. So there is a relationship between the three, but in some senses this is blurred, such as Charlemagne's so-called "imperial coinage" which has provoked a lot of debate. While some historians have seen the coins as representing this Roman heritage, others have also seen other influences. Rosamond McKitterick in her book Charlemagne: the Formation of a European Identity argues that the Carolingians relied on a plethora of earlier traditions in order to augment, maintain, and construct their political power. If you have any other questions let me know, I tried to give more of a broad overview.