It is doable but not, in my opinion, very feasible. You will make six major relocations and it always takes longer to go from Point A to Point B than planned even if things go well. Each relocation will eat up 3/4 of a day (or more) each time. That means you'll lose a great deal of time just relocating. To keep the amount of lost time to a minimum, strive to keep it to only two places (three at the most) and do a day trip or two from each. Perhaps London, Paris (easy to get to from London), and ? For the sake of harmony, have the family decide on the two (or three) places that are of the greatest interest and stick to those.