We are planning to start in Asia and then Europe, Italy and France especially, and then perhaps Season in different places, for 3-6 months at a time to help immerse ourselves into a community and a language and also keep costs down.
Unless you have dual citizenship you are limited to 90 days anywhere in the EU/Schengen Zone every 12 months. They are serious about that. France and Italy are both in the Zone as is most of Europe except Switzerland. Virtually impossible to get a long-term visa in Switzerland as an American without a work reason.
There are other issues with being without a base, but I'm sure you have researched them. Special issues with kids as well.
We lived and worked in Europe with a different type of visa for years before recently returning to the States. We will keep a house here but plan 1-2 month stays in different places each year.