The reason you might not be able to find an answer to this specific question is that it's not really a moodle migration, but a domain name change - if I understand you correctly ...
abcwebsite doesn't need to exist or be running anymore and you want the Moodle you have to be accessed by xyzwebsite, is that correct?
If that is correct, using what tools you have for domains/DNS, change it.
The only thing you'd have to do in Moodle then is to edit the config.php file replacing oldsite with newsite.
And then login to the Moodle and run the search and replace tool ... search for the FULL URL: http://oldsite/[moodle]/ (including trailing slash) and replace with FULL URL: http://newsite/[moodle]/
Include [moodle] above IF you ran your site in a moodle directory.
If I am in-correct, please describe/correct.
'spirit of sharing', Ken