Sure it does!
Aha. So it does! Not looking hard enough, clearly
Given where I am, I've signed up for DropBox so at least this will work for me in the future, but as I've almost completely decommissioned the old phone now, I'm not going to try the synch process for real.
However... I gave Whenlast permission to do its thing with DropBox and sure enough, it's created a folder (apps/whenlast) in DropBox. I've got automatic synch switched on (I've got a decent mobile data plan and I figure this is going to be a relatively small amount of data that's being moved) and Whenlast says it's done its stuff in that respect.
The Whenlast folder in Dropbox was empty until I created a new item -- at that point, a sync folder was created and a file created in it.
So should the process have created a sync file once synchronisation was enabled?
[For what it's worth, the normal export process DOES create files as expected on my new phone. Whether this is down to differences between Lollipop and Nougat, a permissions issue on the old phone or a result of opting into the beta programme, I can't guess.]