All in One WP Migration breaks with Local Lightning beta

I run both Local by Flywheel and Local. When exporting a site from Local, it fails to import on Local by Flywheel. The database becomes somewhat of a mess, and I have to re-install WordPress from scratch.

I have tried all possible combinations of different versions of Local by Flywheel, Local, and the All-in-one WP Migration plugin - still no joy.

As an aside, I even tried using Flywheel’s own export-import mechanism. That did not work, either. I don’t know if it is related, but I found it worth mentioning.