I just performed a site migration a few minutes ago and had no issues.
After first clicking on the ‘Export’ button, you’ll see a dialog that says the files are being prepared. Once that finishes, there’s a ‘flashing’ button that says:
DOWNLOAD [site name]
SIZE: XX MB or whatever
with a CLOSE button underneath that.
It took me a few minutes the first time around to realize that the ‘DOWNLOAD [site name] thing was actually a button and you have to click on it to download the migration file. Could that be what happened in your case? (Once you get it downloaded, or saved to DropBox or whatever, you then install the plugin on the new site and perform the IMPORT on that side.)
Hope this helps.
Terry