My understanding is that the primary reason the papacy was in Avignon for a time is that the popes of the era were clients of the Kings of France, but I'm unclear on whether Avignon was part of France at the time.
If Avignon was not controlled by the Kings of France, why was it advantageous for them to keep the Popes there instead of in Rome (or somewhere actually in France)?
I've tried to figure this out on Wikipedia but different pages have conflicting answers (as well as dubious sourcing). The most coherent I've seen has Avignon in the Kingdom of Arles, within the Holy Roman Empire, during this period.
Avignon was formally part of the Kingdom of Arles when the various popes lived there. Arles was independent from France and remained formally a part of the Holy Roman Empire. It was culturally distinct from the north in many ways. However, despite the formal distinction of the region, it was basically settled that the King of France controlled the region, even if not legally. The last pope to try to claim power over the King of France by excommunicating him (Pope Boniface VIII) wound up getting beaten in his home by French and Italian troops that the king had sent, which eventually led to his death. The next pope decided to absolve the King of France, and after him, the next pope (and the first pope to reside in Avignon) was a personal friend of the king. The reason Avignon was chosen over Rome was due to a similar conflict between the church and the Holy Roman Emperor, both of whom were vying for secular and temporal control. Avignon was seen as a safer area for the church, particularly due to the positive relationship with the King of France. The area surrounding Avignon was already owned by the church, and in 1348, the church purchased Avignon and united the two areas.