This organization you described is not the WordPress standard. Usually when there are multiple WordPress installations within the same database, a prefix is used for each installation. Here you can identify the tables by the prefix and export them separately.
If you still have access to WordPress from each client it is best to use the WordPress exporter to move the sites. It will take care of the database and all attachments and you will not need to worry about how the database was organized.
Depending on the size of the site and server performance, the import may give timeout or memory error. In this case you have two options. Either separate the export xml into smaller pieces or use a backup plugin to transfer the database.
If you are going to use a backup plugin, I suggest the BackupWordPress . You need PHP greater than 5.3 to run. It only backs up the database so it will create a sql file only with the tables relative to that site.