The thing to remember about The Soul Rebels sound is its deep roots within Afro-Cuban music. Not only are the percussive rhythms present throughout their music, but it could also explain the syncopated bass lines and minor key tonalities. What seems to be lacking from most other brass bands outside of New Orleans is the direct connection to the Afro-Cuban roots.
The Soul Rebels have carried these traditions into the modern sound by incorporating funk and R&B elements, making use of extended chord voicings and scales. Within the minor key tonalities, the use of harmonic and melodic minor scales add tension to the sound that wants to resolve. But when coupled with syncopated rhythms, it can stir the listener in to a frenzy. Also note that the chord progression changes during the solo sections.
The trombone solo is played by Paul Robertson, and one element to make note of is how flexible his sense of timing is. While the pulse is a marching funk, Robertson's phrasing can be very relaxed at times, almost behind the beat. This laid-back style is similar to Caribbean trombone where the music may be coming from the beach and not from the street parades. Another Caribbean technique that Robertson tends to use is the scoop into a note, adding a slight glissando to bend up to the desired note, as well as adding a slide vibrato to longer notes.
Here's a YouTube video the band showing off some dance moves for "Turn It Up" during the trombone solo.
Recommended Reading: Volume 57 - Minor Blues In All Keys by Jamey Aebersold. Published by Jamey Aebersold Jazz.