It this is is more than a handful of sites, I’d handle the migration from the server level, rather than exporting and importing individual sites (and dealing with media issues).
Here’s a high-level overview:
1) Dump all the MySQL databases, tarball all the individual sites’ files, and SCP everything to the already prepared new server.
2) Import the individual databases into the MySQL server on the new server, and untar and move the individual WordPress sites’ files to appropriate locations.
3) Point the domains to the new server, and perform all the obligatory sysadmin dances to ensure everything works fine.
This, of course, requires the right level of sysadmin skills and confidence to pull off… especially if something doesn’t work as expected.
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If you still prefer to manually move one site after another, here’s one little trick which is definitely quicker, easier, and safer than using the WordPress IMPORT and EXPORT function: use the free ManageWP service!
ManageWP is a central WordPress management platform (acquired by GoDaddy a while back). The basic features are free for an unlimited number of sites, and, surprisingly enough, that includes one free backup every month which is stored on their servers. And it’s this free backup stored on GoDaddy’s servers that you’re going to leverage to do your migration.
1) Install and activate the ManageWP Worker plugin on a site, connect the site to the ManageWP dashboard, and run a backup.
2) Point the domain (of the site you just backed up on ManageWP) to the new, already-prepared server, run a fresh WordPress installation there.
3) Install and activate the ManageWP Worker plugin on the newly-installed WordPress site, connect to the ManageWP dashboard once again, and RESTORE the backup.
I recently used this process to move a large-ish site and it went very smoothly.
I’d suggest you give this a try with one site!