Thanks for the further details 

A couple of things that will be helpful for this is - there is a report source 'Assignment submissions' that can be be scheduled to be sent to the trainers with a saved search for courses required. 

There is a setting in the assignment under Grades called workflow setting - when this is enabled there is another setting Use marking allocation.

This will allow trainers to be allocated to specific users submissions - once allocated they can be marked as required or reallocated.

Let me know if this sounds suitable.

Regards

Craig