What is the best way to transport a large quantity of LEGO bricks?
When I moved house, I just sealed the gaps in a bunch of packing boxes with duct/packing tape, and put the storage bins in there - that way if anything did escape from the bins it wasn't going to get lost from the box.

I would say though that in general my collection wasn't organised in anyway, although there were a few built sets that survived fairly well.