We use import/export XML script to migration content at the moment.
http://manuals.matrix.squizsuite.net/server-administrator/chapters/import-scripts#import_from_xml.php
But those scripts are just helping around to move content. It doesn't move implementation like triggers, server specific conditions, system assets, and any custom implementation depending on urls, server specifics or sometime some hard coded url, javascript, asset id etc. There is no guaranteed and it's just impossible to make sure it can automatically move a web site to another instance of Matrix, and instantly working perfectly.
It will still involve manual effort to check, set up or fix implementations that got moved by the script.
It is best to talk to Squiz support and get them to estimate your project.