I know that the idea of fixed borders is a modern one, and that in the past they were much more fluid. What was life like for the people that lived in in areas which were contested by various counties/duchies/kingdoms in the middle ages? Who would they pay taxes or swear fealty to?
First of all you have to specify which part of the middle ages you mean. There is the early middle ages, the high middle ages and the late middle ages.
In the first one the most important kingdom was the Carolingian empire. This started out as a rather centralized empire, but only for the elite. They lived in a working feudal society, where the dukes and counts and the lesser nobility where bound by oaths to the king/emperor and eachother. This system worked because Charlemagne kept his nobility close, he needed them to fight in his army. The common people however was still used to live in the tribelike structures they had known since the arrival of the Franks, Burgundians and other tribes. The years passed and the nobility returned home to rule over their lands. The connection with the kings became more and more distant and the lords became absolute rulers over their own domains. Even most of the lesser lords stopped listening to the count or duke the swore fealthy to. They had their own little army of knights and started fighting little wars against each other. This was bad news for the common people. They already had to work for their own lord and now there was always the possibility that some other lord came by and took the rest. This period is called the Feudal Revolution, but there is still a debate going over this term.
From the late 12th century some stronger rulers rose to power. these centuries are called the High middle ages. The stonger lords brought the lesser lords back under their rule and started making war against eachother. Some of the contested territories are: the whole of France (100 years war, Engeland vs "France"), Flanders and Scotland (both regions had a strong local nobillity that came in rebellion against their king (the French one and the English one), Zealand (count of Flanders vs duke of Hainaut/count of Holland), the christians kings vs the muslims in Spain,... Most of these conflicts are entertwined with each other (hundredyears war, Flanders, Zeeland for example). For the common people this meant: being draft to fight in the army of there lord (this mostly meant getting a piece of wood with an iron point to fight against some iron man racing his enormous warhorse in your direction, in some specific situations this was a different, but then I would be going in too much detail.). It also meant that they had to feed those armies passing by, sometimes with their consent, most of the times not. This also meant that if you had daughters, you beter had to hide them... Meanwhile they still had to work for their lord. At this time the trade with asia started to go at a higher speed, unfortunately the merchants also brought the plague back. And we all know how that worked out.
I won't talk about the Late middle ages, because for the common people it was kind of the same thing, but with some better equipment.
When I talk about common people I mean most of the people who lived on the countryside. Those who lived in city's where less affected.
I apologize for my English, I'm not a native English speaker.