If you want MusicBee to behave like iTunes, you should enable the auto-organize feature in Preferences > Library. This page may help: http://musicbee.wikia.com/wiki/Templates
Sorry to contradict again, but I wouldn't recommend that to someone already having troubles with their library - it makes significant changes and cannot be undone.
I believe the iTunes behaviour requested is the ability to move an entire library (including music files) from one location to another, not the ability to have MusicBee name and organise files on-the-fly automatically.
This is easy enough if all of the music lives in a single parent folder, but how do you "move" a library that could include tracks from any number of different folders or drives?
I'm not about to install iTunes to see how it handles it, but some quick googling shows that:
1. I was incorrect earlier when I suggested iTunes forces users to keep all of tracks in a single folder (although it will by default)
2. iTunes includes a command to consolidate
a library so that all of the tracks are in a single folder. From there, it's a doddle to move everything at once to another drive.
In short, this is probably something that should be made easier in MusicBee. If only because of the number of people who have posted questions after trying to do it.
Bee excellent to each other...