It was easy!
Ok - I did have help from cafuego, so there's some bits of the process that were just magic to me. But once I had access to the environments, I enabled the migrate modules, and followed instructions on drupal.org and hey presto it was done!
I did need to do manual clean up - re position blocks into differently named regions, and recreate some menu items for taxonomy.
I also got stumped for a bit about why I could no longer free tag, but that turned out to be a simple setting change.