We were there several years ago, we think on a Sunday, but can't be sure. We were in the main square and a procession of students? made their way along the square carrying flags and playing music. Is this something that happens frequently or were we just lucky that day?
Google: Siena processions or parades. Are you trying to catch another procession in another city?
You were probably lucky but I can’t give you a definitive list. The main one is around the horse race (Palio) and there is another at Easter. There probably are others.
Late June, getting ready for the Palio, in medieval dress?
Could be anything including a protest. The last time we were in Madrid we walked into a protest march by medical students. Quite large and noisy. Lots of police around, sign carriers, etc. Who what you saw.
Best we can remember if was in the Fall, and it looked mostly young people - carrying flags and banging on drums. Thought perhaps they were from one of the churches or schools. It was a pretty big procession or parade. Didn't know if this was something they may do often.