Hi Istiophorus
This has to be one of the most asked questions on Wetpixel! And there have to be dozens of solutions with everyone having their favourites.
The solution, I believe, if you want to move all that gear in one one box is called "a house". 
Pelicans cases - or similar - are great. They sure take the knocks, protect the gear really well and will last for ever. The downsides are that they are heavy and expensive. If you want to take one on a flight you need to check that the one you chose is within the limits of the airline cabin baggage allowance. I think the Pelican 1510 roller box is about the limit. You should be able to get a housing, strobes and EXRs into that - but probably not, say, an 8" dome port as well. But DO check carefully as Ikelite housings can be quite big.
And watch the weight! A fully loaded Pelican case can easily weigh 15kgs. Let's not discuss how you get that into the cabin inside the usual 7-10kg limit.....
I haul a D800 housing, a 9" dome, 2 Inons strobes, a flat port and then all the usual bits and pieces by air; The housing, flat port and strobes go in my Pelican case as cabin baggage (no problem so far - 15 years on); the 9" dome in a padded out cool bag hand-carried; and the rest (arms, clamps etc) in my hold-loaded bag with my dive gear. I also have a softish camera bag with me which has 2x D800s and lenses.
If you want something to haul your gear around at the location, a cool bag works well; and my own personal favourite, the humble laundry basket. Self-draining, cheap, disposable, buy one at any hardware store or supermarket, donate it when you leave, available in a range of colours. Stick a bin bag inside and you also have your own rinse tank - great on dive boats with small rinse tanks or where people happily throw their mask and snorkel in the camera rinse tank.
So, there's a couple of ideas..... they work for me.
Richard suggested the ThinkTank Photo Airport Series. They sure look great. Toss up between that an a Pelican I'd suggest. Advantage of the Pelican is that if you have to hold load the bag (God forbid), you might worry less with a chunky Pelican case over the softer ThinkTank. But for hauling gear around, the ThinkTanks looks terrific.
But you have to face it, there is no perfect solution: we all chase one. We never find it.... a bit like the Holy Grail. 