Hello,
There are a lots of issues when migrating sites, if talking about client's migrations:
1. PHP version incompatibility, extensions incompatibility, or missing extensions
2. Issues with importing databases, sometime huge databases can only be imported through CLI, some databases require manual modifications in encoding. Some databases are messed up.
3. Always exists issues when migrating sites with unupdated CMS (like joomla 1.5 from early 2010, that's a headache)
4. Email deliverability after migration.
5. some sites which worked on apache and are migrated on server with nginx+fastcgi or nginx-cgi require rewriting of htaccess to nginx configuration.
and more and more...
I can't count how many hours of troubleshooting have been done for all of this.