Is there a technical name for a several mile stretch of flight before landing?
The best term to describe it would be the 'final approach segment', which is defined by ICAO as:


  This is the segment in which alignment and descent for landing are made. Final approach may be made to a runway for a straight-in landing, or to an aerodrome for a visual manoeuvre. 


The FAA defines the same as:


  The FAS begins at the PFAF and ends at the landing threshold point (LTP).  The FAS is typically aligned with the runway centerline extended. The segment OEA is divided into the FAS OCS and the visual segment obstacle identification surface (OIS). 


Note: Both Eurocontrol (which refers to the ICAO document) and Jeppesen define it only for instrument approaches.