If only your source and target domains are different (but your usernames are the same), the quickest thing is to set up a manual Domain Mapping in Content Matrix.
However, if your source and target usernames are also different, then there is no way for Content Matrix to auto-map, since it doesn't know what source and target users to map.
Hence, the only solution is to set up a manual mapping for each user, using a user mappings xml file, and then importing that file into Content Matrix.
Please find instructions below:
https://support.quest.com/technical-documents/metalogix-content-matrix/8.7/sharepoint-edition-user-guide/30#TOPIC-1059343