My specific area of expertise is that of the Austrian Habsburg Monarchy, so I will focus on them. I'm afraid I can't offer more general comment as to how Brandenburg-Prussia or another Imperial Estate would have administered its territories outside of the Empire, but hopefully another flair with a wider array of experience than me can come along to try and examine it. Now, of course, questions involving the HRE involve a time period of nearly a thousand years, and of course, things will be different if we're discussing the Holy Roman Empire of the Ottonians in the 10th century, or the Habsburgs in the 16th. Again, my experience is limited, so I will primarily be talking about how the Austrian Habsburgs organised their realm in the early modern period, primarily looking at the latter half of the 17th century, after the conclusion of the Thirty Years War.

Looking at a map of the Austrian Habsburg Monarchy we can see that the core territories of the monarchy within the empire are the Archduchy of Austria itself and the Kingdom of Bohemia (each of which in turn consisted of a large number of constituent territories). Perhaps the key distinction between territories within and without the Empire was the combination of loyalty to the Emperor, as well as a certain number of rights and privileges accorded to Princes of the Empire under the imperial constitution that would not be granted to non-Imperial estates. I've gone into more detail on the differences between imperial and non-Imperial territories here. At the highest levels, the Habsburgs effectively maintained two distinct organs for diplomacy, one for policy within the Empire and another for diplomacy without.

Outside the boundaries of the Holy Roman Empire, the Habsburgs also claimed the title of King of Hungary, though prior to the Great Turkish War, Habsburg rule was limited to the relatively narrow stretch of land termed Royal Hungary. Yet, how did we arrive at this situation? Unsurprisingly for the House of Habsburg, their claims to the Kingdom of Hungary rested on diplomacy. In 1526, Hungary and Bohemia were both ruled by King Louis II of Hungary, a member of the House of Jagiellon. An important thing to remember is that, much like the HRE as a whole, both the Kingdom of Hungary and the Kingdom of Bohemia were elective monarchies. Kingship did not automatically pass to the eldest son, but rather, on the death of a King, the assembled nobles of the Kingdom would elect a new King. Normally, of course, this would be the son of the current King, but--much like in the Holy Roman Empire--the politicking to ensure the election of the desired candidate provided the nobility a powerful balance against the authority of the monarch. This was a fiercely guarded privilege and the claims of Bohemian nobility to elect their own ruler was a key step that led to the outbreak of the Thirty Years War, but I'm going off on a bit of a tangent. Back to King Louis II and 1526.

In 1526, King Louis II was killed in the Battle of Mohacs, waged against the Ottoman Empire. Louis II died childless, and had no clear heir. To add to the confusion, the defeat at Mohacs had seen the main Hungarian army completely destroyed, and the Ottomans soon marched on the capital of Buda. Prior to the conflict, there had been a serious divide in the Hungarian nobility, between a pro-Habsburg part centred around many of the senior nobility and clergy, and an anti-Habsburg part focused around lesser nobility and Protestants. These two groups would go on to elect two different Kings of Hungary. The high nobility would give the crown to Ferdinand of Austria (younger brother of Charles V and future Holy Roman Emperor), while the lesser nobility would do the same for the Voivode of Transylvania, John Zapolya. Eventually, Hungary would be partitioned in three, with the Habsburgs taking a strip of land along their own frontiers that would come to be known as Royal Hungary, the Ottomans occupied central Hungary, while the rival Kings of Hungary would ultimately consolidate their power base into a new Principality of Transylvania.

As far as administration, Habsburg rule over Hungary and Croatia was indeed quite different from their own rule over their own Austrian or even Bohemian territories, but as much of this was due to the decentralised nature of early modern administration as a whole. It's important to remember that most early modern polities were not unified states, despite how they appear on the map. Many are what we would call composite monarchies, with a variety of different territories (each which their own sets of traditions and laws) united under the rule of a single monarch. This distinction is highly important when we consider Habsburg rule over non-Imperial territories, as--in Hungary--the Habsburgs did not rule as Holy Roman Emperor, but rather as King of Hungary. As to the administration of these realms, rule of an early modern polity is far different from what we might imagine today. There was no central bureaucracy responsible for promulgating and enforcing decisions made by the head of state. Rather, enforcement of royal (or even Imperial) decrees was heavily reliant on cooperation with local nobles and other such individuals (usually called the Provincial Estates) who would actually be able to do whatever it was the Emperor in Vienna had asked of them. Thus, Habsburg rule was a constant balancing act between the Provincial Estates and the Emperor as to how much would be done. To give an example, while it might be quite easy to convince the Provincial Estates to vote for taxes meant to raise an army to defend their lands, it would be much harder to convince them to do the same for an aggressive campaign that might not involve them. This was as true for territories outside the Empire as it was within. So, to summarise, it was hardly surprising to see a non-Imperial territory be ruled by a Prince of the Holy Roman Empire. However, the importance of the composite monarchy and the difficulties in realm wide administration cannot be overstated. Hopefully, this has helped to answer your question.

Please feel free to ask any follow ups.