The country seems to be ruled by two competent old ladies having fun in their job. Nobody under the age of 75 seems to work there, as they are all out of the country, skiing, sleeping, spending time in France (what is Marie doing there?) and doing some military training that apparently is very important for Denmark.
Marie has events for both her French and Danish patronages in France - they're just not always covered by the Court (for example, a visit to UNESCO as patron in October or the Sidaction gala dinner that she attended as Patron of the Danish AIDS Foundation last month were not announced or covered by the Court at all IIRC). She also flies back to Denmark to have events. All in all, despite moving to France, she probably works as much as she used to while living in Denmark, it's just that her activity is concentrated on 2-5 consecutive days a month rather than 1/2 days a week. It makes sense to me that she would move with Joachim as he undertakes this training as a natural continuation of his military job. Let's keep in mind that soon enough Joachim & Marie will not be needed as Christian & Isabella will take over and I find it remarkable that Joachim is getting ready for this. This training is perfect for him to get a NATO job later on for example.
As for Fred & Mary, that's another story and I still don't understand why their stay in Switzerland was necessary. I could understand why they'd want the kids to study abroad but it's the timing and the way it's done that puzzles me.