We made 3 mid-year moves when my kids were younger. In each case, it took a good 2 months for them to catch up with the rest of the class (you would think they would be ahead in some areas, but honestly, learning a new teacher's style, new test processes, new classroom rules... it canceled out any advantage they might have had).
Schools teach different items, at different times, using different tools and resources. The worst was math, as the curriculum, subject matter, method of teaching, etc. was extremely different between school districts (and between states it wasn't even remotely the same). Science and history are both taught at very different times, so one school may be covering early American history but the other is covering world history. My daughter skipped biology topics, but covered earth sciences twice one year.
Honestly, it is a nightmare switching mid-year. Switching twice in a year? Every year? I would predict that kids in that situation would be perpetually behind their peers in learning.
Once the children are in high school and their classes go on a transcript, they would end up with a lot of "incomplete" courses and that would significantly impact college options...
I think the families that do this successfully either homeschool, or enroll their children in an online school (which is just as parental-labor intensive as homeschooling).