How do they do it? They source wrecked Leafs and harvest the battery and parts.
When I bought the LEAF, I paid $800 from San Diego to Memphis. It all depends on space availability, routes (are they already going that way) and how much they want your business. You go through brokers they take a commission. Shopping around helps. I had quotes of $1000. The car arrived on time in great condition. It cost more for enclosed trailers of course. I did about 1000 mile trip last month (in my VW TDI). I saw many car carriers with used cars....In this scenario $9000 vs $3000 for battery swap I can afford to ship the car.
My thought as well: if they can manage the battery pack cost for me (not to mention the hassle), then I might be able to justify shipping my car.
Since you're the OP, you should be able to go back and correct the "battery" typo in the title