" I know that Sousa bands and the bands of that time period used cornets, and I know that later big jazz bands used trumpets, so the transition probably occurred some time in that period in the very early 1900s".
Here is an alternative explanation to the novel story of Bob Pucci's!
Although Halary added valves to the post-horn & thereby invented the cornet, it was Sax who created a "family" of cornets - i.e. the saxhorns. The over-the-shoulder American Brass Band (despite the coiling) is a true saxhorn ensemble. The so-called British Brass Band comprises a mixture of cylindrical instruments (the trombones), Tubas (rapidly flaring conical horns - the BBb Bass, Eb Bass, Eupho & Flugel), and saxhorns (gently flaring conical horns - the Baritone, Tenor Horns, Bb Cornets and Eb Cornet).
The evolution of the "Jazz" big band has been traced from ty the progressive replacement of the instruments of the traditional Brass Band in New Orleans. First the Eb Cornet was replaced by the Eb Clarinet (later replaced by Bb clarinet), the Tenor Horns (peck horns) were replaced by the Alto Sax, the Baritone by the tenor sax, and finally the Bass replaced by the baritone Sax and string bass.
Though cornet started life as an instrument of about the present length. The trumpet gradually evolved from the standard natural trumpet of twice the present length. There were four evolutionary variants in the attempts to chromaticise the trumpet, rotary valves, piston valves, keys, and slides, but the instruments remained long (G, F or Eb ). The trumpets played the traditional repertoire, and were given modified variations of it by the contemporary composers, whereas the "NEW" cornets were given a rather different type of part . The Paris Conservatoire had seperate departments for Cornet & Trumpet, and a certain rivalry arose between trumpeters (descended from the heroic military trumpeters) and the cornetists (virtuosos or dance hall musicians, depending on one's perspective). The shorter Bb trumpet first appeared in Germany around 1870,and in about 1880 Besson came up with the design of what has become the accepted modern piston valved trumpet. With its tapered lead pipe and bell flare plus shorter length, there was considerably less cylindrical tubing than on the older double length trumpet, but it was not (and has never become) a truly conical instrument.
The Bb trumpet was first introduced to London Orchestras as late as 1912 (by Ernest Hall). The transition from cornets to trumpets of the same length followed in all areas (except the Brass Band!), to the point where the Orchestral use of the Cornet became frowned upon, even for performing Cornet parts!