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 running on old server, while simultaneously creating site 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 to that way the dns can still point to the old server to grab the media files. Then create a new root site for and point the DNS to the new server, then migrate again from back to just

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

Any ideas? TIA