What is the best practice for mapping from natural keys to integer-based keys? (ETL)
I have found building a relationship table between the old (natural) and new (integer) keys works well in conversion processes such as this.  The integer key is built using the standard auto-increment method for the database.  Maintain unique keys on the data which forms the natural key in the new tables.

Develop a repeatable data synchronization method that can be applied as often as required.  Depending on your needs this can be daily, weekly, monthly, or on-demand.  Make the synchronization modular so it can be updated as required.  In some development processes you may end up feeding from the new system back to the old system.

As you develop and deploy the system, track which system is the system of record for which data.  Move control to the new system as possible.  You may want or need processes to identify and deal with changes which occur on the system which is not the system of record.

Manage change overs from the old system to the new.  If possible, disable updates in the old system as the system of record is changed to the new system.  This will require a feed from the new system to the old.