Hi you will need to have a filter on the choice list for districts and related data. Please see the below post for examples. Filter based on previous choices - #7 by Grzesiek2010
In your example though for this to work the enumerators will still have to select their name from a previous select_one question. I don't think that there is currently a way to retrieve the username from the server login unless someone can correct me on that. The choices available in the district question would then all be filtered based on the enumerator name.
A second option would be to give the enumerators ID cards with barcodes which they scan and which automatically fill these questions using the pulldata() function.
Noel