I see you planned a similar trip to Italy (sans kids?) for December 2018. Did you stick to the plan and have short stays in several places? How did that go?
Barcelona and Paris both have a lot to offer and day trips too. Especially for kids, I think Barcelona will be more interesting. From Paris to Barcelona you can spend 6.5 hours on a comfortable train or you can save a couple hours by flying 2 hours (you'll need extra time to/from airports, plus lead time at the Paris airport, plus wait time in Barcelona if you are checking bags). Either way you use most of a day in travel. You'll use another 1/2 day to change locations for each additional place you go to.
Unlike moving around a single country like Italy, you'll have multiple languages to interpret, and with London another currency to figure out. Moving 4 people usually takes longer than moving 2. Remember, you will never move faster than whoever is slowest at any given time (sleepy head, window-shopper, picture-taker. . . ). Added time if you're 4 people in a room with one bathroom . . .
It's our nature to want to pack in as much as possible. That doesn't have to mean lots of different cities, it can mean lots of different experiences in fewer places.