I have now successfully completed both my written and oral comprehensive exams, and am one large step closer to finishing a doctorate in multiple woodwinds performance.
In the oral exam, one of my professors asked why woodwind doubling is a well-recognized musical specialty, but doubling on brass instruments is not. The question was an odd one, especially since brass instruments fall precisely outside my area of expertise. I didn’t have a good answer, except that brass players seem to be particularly protective of their embouchures, and presumably don’t want to risk ruining them by switching instruments. (That seemed to be satisfactory for purposes of the exam.)
I do know of one school that offers a “woodwind specialist” master’s degree, a “string specialist” master’s degree, and, yes, a “brass specialist” master’s degree: Michigan State University. (Degree descriptions here.) The string degree requires one primary instrument and one secondary, the brass degree requires one primary and two secondaries, and the woodwind degree requires one primary and three secondary instruments. I expect if anyone is doing the string degree, they do violin/viola or cello/bass, hoping to get one of the “high strings” or “low strings” teaching jobs. The only combination of three brass instruments that strikes me as marketable is trombone/euphonium/tuba, a “low brass” specialist.