Moving single site from Multisite to another Multisite server

Okay, here's the situation. We have one multisite, multi-network deployment with a LOT of sites. I've got another fresh multisite, multi-network install that I'm going to peel a few sites back to.

The duplicator plugin doesn't seem to be a good fit since I'm not looking to backup/restore the entire database, just a few sites. I've been experimenting with the Wordpress Importer, and it seems like it will do what I need it to do, but here's the catch.

How can I have xyz.com running on old server, while simultaneously creating site xyz.com on new server and have the WordPress importer grab the media from the old server. I can only point the DNS one way, either old or new!

I've never had very good luck with renaming a site after the fact, and it always seems pretty messy. Even on this fresh 3.9 install I'm seeing some weird behavior when trying to move a site from one network to another. Plus I can never seem to get a site to take over the root site (/) of the destination network.

I thought about doing a double migration - ie move xyz.com to abc.com/xyzsite that way the dns can still point to the old server to grab the media files. Then create a new root site for xyz.com and point the DNS to the new server, then migrate again from abc.com/xyzsite back to just xyz.com

Obviously using domain mapping plugin would be an easy workaround, just mapping abc.com/xyzsite to xyzsite.com, but I would not end up with xyzsite.com being it's own network with its own subdirectory sites (it would still be in abc.com network).

Any ideas? TIA