Does it copy all files in ~/Library ? Including cache files and even things like Flash cookies? What about user created files/folders in / ?
Migration Assistant is fairly well documented in http://support.apple.com/kb/HT4413 - basically if a newer or equivalent version of a file is not present, it gets moved by default.
In your case ~/Library is moved if you select that user - you can't deselect it like some other folders in the home folder.
Applications (which includes /Library/Application support as well as /Applications) move when that is selected.
Files in / only move if you choose other files. Everything goes AFAIK. Even ByHost preferences that no longer are used since the host has changed.
The Migration Assistant will write to the logs, like every other application and service. Please see your logs for these details: