If you are looking down the line at moving, your job priorities will probably dictate when the time off comes. But if you have some flexibility in the timing, I recommend choosing to go at the very end of the summer with just enough time to settle in and adjust to the time change before school starts. For us that is a week, for others, maybe longer.
This way, you manage to capitalize on the off-season prices of homes, in Europe at least. By booking long term in the long off season, Sept 1 through June 30th, you can find a home owner who is glad to have their home filled securely for the whole off season and can offer you an excellent price. This can be 25% of the high season summer prices in our area.
Also, if you are moving children in to a school system, the easiest thing for them is to start at the beginning when the others do, and stay with consistent schooling during the whole year. And then what to do over the summer time? As you wish, either stay where you are, hopefully at a discount, move around and travel or explore (see what we did at www.roamingfrance.com/6-weeks-6-countries-3-kids-1-tent). Or head home to enjoy your own high season.