As an example, a traditional way to harmonize a melody in top-down, close-position writing for 5 saxophones is the 1st alto plays the melody note, the 2nd alto plays the chord tone below the melody, the 1st tenor plays the next chord tone (below that of the 2nd alto), the 2nd tenor plays the next chord tone, and finally the baritone doubles the 1st alto an octave lower. While there are 5 saxophones, the chord voicing is actually 4-part harmony.
Top-down, close-position 4-part harmony as described above for saxophones is also used for the brass section. The 4 trumpet parts are usually written first and then the 4 trombones often double the trumpet parts an octave lower or with variations of this technique.
Open or spread voicings can go beyond the 4-part harmony used in close-position writing. In spread vocings, the trombones and/or lower saxophones often play basic chord tones (such as 1-7-3) and the trumpets and/or higher saxophones play chord tones or tensions. Including tensions (natural or altered 9, 11, or 13) in a voicing gives a greater amount of richness to the ensemble sound. Spread voicings can use 5, 6, or 7-part harmony instead of being limited to 4-part. However, unison and octave doubling still occur when even 7-part harmony is used in concerted saxophone & brass passages.
Given this perspective of harmonic doubling in big band music, I have an idea to offer to big bands: Try playing my mid-size ensemble scores with doubled players on a part. As I envision this approach, it can give a fresh sound to conventional big bands.
Bands are encouraged to experiment with this idea. Of prime importance is having lines with individual tone colors instead of harmonized saxophone, trumpet, and trombone sections. To do this, an expanded palette of tone color is created by using the brass in combinations of different mutes & open with saxophones and woodwind doubles. Including instruments such as soprano sax, flugelhorn, bass clarinet, and tuba is a good way to add more color if they are available as doubles in an ensemble. With this palette of color choices, each wind instrument is taken out of its customary role in a section and combined with another instrument(s) to produce a distinctive sound for a given line in the score. Creative combinations of instruments are encouraged.
As a way to begin, I recommend my piece Fanfare for big bands to try. It is reasonably easy to play (can be performed by high school, college, professional and community ensembles) and its page on my website has a suggested assignment of parts for a big band.