It sounds very exciting, and I really hope we can take the family travelling like that one day!
I would think twice about having a base and doing long 'weekend' trips from there, unless you have long term accommodation in your base, and a lot more stuff than you plan to take with you for the long weekends. But if you just have what you can take on the plane, and don't acquire heaps more there, you may as well load all your stuff in the car and do a slow circle, stopping in each place for several days, and you cut out masses of boring travel time which lands you back at the same place again. Europe may have lots of countries, but it isn't really that small, and it can take a long time - a full day or more of travel, to get from one place to another, eg, Lyon to Barcelona, Lyon to Prague. Motorways and train lines in Europe are not always scenic and interesting, and I think with kids I would be aiming to minimise the number of long travel days.
