I have been a WPMUDEV customer for some while and recently migrated my first site from an old, slow host to a Bronze WPMUDEV hosting package.
I used Shipper for the migration process and used the Import method.
My site has been connected to the WPMUDEV Hub for some months and I make use of many of the plugins, including Hummingbird.
After the Shipper migration had finished, most of the site looked OK at first glance, but then I noticed there were a ton of JS errors showing up in the browser console which had not been there with my previous host. This caused some issues with other plugins.
A conversation with WPMUDEV support showed that the JS issues were caused by the Hummingbird cache being imported to the new hosting. The support guy, Luis, told me this is a common issue and he advised me to clear the Hummingbird cache.
Once this cache was cleared, everything started working correctly and the JS errors disappeared.
Since the site was connected to the Hub both before and after the migration, and it is possible for Shipper to identify that Hummingbird was in use, perhaps you could modify Shipper to automatically detect and clear the Hummingbird cache as a final step of migration?
This would result in a much cleaner migration.