I view this as an adventure that is not really about camping but just a young family moving around without a car. The actual tent bit that's probably the least part of the problems. Kids get tired+sleep, they don't demand comfort like adults. However kids have nil tolerance for boredom and make their own fun which may not be convenient for the situation.
So, I doubt this will be done but you can actually do these type of adventures by folding bikes with trailers. The bike part resolves the issues of moving children and luggage, the folding bike part solves issues of fitting the stuff on trains/boats/planes/cars/etc. I don't have photos to show the exact solution but you can extrapolate.
So here is an example of the packages on a train, I have similar photos at airports.
Here is a bike with a large trailer, in an appropriate outdoors camping setting (Langdale)
and here is a large tent which you can change to one right size for a family of 5, e.g an Hillberg. The trailer i used had more camping-specific gear than you will use and I was packing for winter.
Here is a bike with room at the front for a toddler.
So I think you can do this with 2 bikes and 2 trailers. one trailer has the luggage (tents, etc) and the other has two kids on tow. One rider then has a a kid also in front. The trailer shown has a 100L and a 40Kg limit, if you wanted more capacity you can add front panniers, rack boxes etc. With this gear, you can fit in trains and boats and parents supply all the forward energy. Once unloaded, you can use the bikes to cycle around, and you can do stuff like drop the saddle and walk the bike with a 5 year old on the saddle.
Once you own the gear, can be used for general stuff, e.g. commuting to work, shopping, etc. Basically you've made your own flat-pack parent-powered family transport system. Cost.... well less than a car, but you're talking about 3 grand I reckon, allow a grand each for the bike, 500 quid for trailers and 500 quid for camping gear. Could do it for less, say two grand but not significantly less.