Once users copy data through a Lookup field, changes to values in the datasource app do not propagate to copied data in the Lookup field.
To update existing records, it is more efficient to update them in bulk by importing a CSV file.
If you bulk update values in the Lookup field by importing a CSV file, values copied by Field Mappings are also bulk updated.

However, following conditions have to be met.

• In the Datasource app, there are no duplicate values in the field specified as Key Field.
• Prohibit duplicate values is enabled in the setting of the field specified as Key Field.

For details on how to bulk update the values of a Lookup field using a CSV file, refer to Is it possible to import values from a file into a "Lookup" field?