I would probably do both. In fact, I did do both when I moved to the other side of the Öresund to southern Sweden a couple years ago.

I packed and shipped two bikes I cared about with some of the rest of my stuff, mostly tools and sentimental stuff. The cheap shipping from California involved a trip through Panama, so I bought a cheap Finnish city bike when I arrived, and I've upgraded that in the intervening years, but retained it's junky look.

Of course, my position was a permanent contract and I came without the extra baggage of family. Since my contract included a capped actual-charge for the move, and some of the perhapses and maybes that I heard about getting some cash when I got here didn't pan out, I probably would've brought my modern MTB and some other stuff, too.