I think this pertains more to music theory than to the Jazz forum so here ya go:

When you're writing a piece for a big band, how would you harmonize 4 of the same instruments? Say, for instance, there are four trombones. Would they all play different parts or would some play the same as each other?

And if they do play different parts how would you go about making those? Like are there flat 3rds or sharp 6ths or something I should know about that are used mainly in jazz?
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it depends on the sound you're going for. Certain charts have most instruments playing melody while say, the lead trumpet and sax doing harmony. or vice versa. or one sections plays the melody and another harmonizes. Or One section has four part harmony and another section has an entirely different rhythmic part and harmonization. there aren't really rules, its whatever fits your sound.

Harmonize maj7's, dom7, 9th's and so on. it might sound jazzier if you have the 7th's or 9th's included in your compositional melody, in addition to the standar b3's, maj3's, fifths and so on.
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That is a huge question, far more broad than anything that can be answered here.

The short answer will sound sarcastic and jaded, but would be "take a course in tonal harmony to find out."

Yes, the answer is a whole course that would involve weeks and weeks of study.

And while you're at it, a course on orchestration wouldn't go amiss either.

Could I get some more talent in the monitors, please?

I know it sounds crazy, but try to learn to inhale your voice. www.thebelcantotechnique.com

Chris is the king of relating music things to other objects in real life.