Take a setting like Cyberpunk 2020, Ghosts in the Shell, Deus Ex, netrunner, etc.
>how does warfare work on such settings?
>what military trends that actually exist now would be more favored then?
>which ones would fall down the wayside?
>would such settings even go to war when a significant part of your population can download themselves in a new body?
>what kinds of weapons would you see?
But OP, the whole focus of cyberpunk is that corporations become so strong that they supplant nation states as the prime denomination of sovereignty.
Governments are weak and ineffectual.
As such, wars don't really happen. It's sabotage and espionage in order to destroy profits.
>would such settings even go to war when a significant part of your population can download themselves in a new body?
yes because taking and holding territory is still an thing and transmitting minds and rebuilding bodies takes energy and material which are limited and can be destroyed by enemy action
but even corporations/estates would want resources that might be rare in their immediate area but abundant in areas belonging to other corporations, what happens when 2 equally strong corporations go head to head with each other.
War probably wouldn't look much different than it does now, save with even more use of remote technologies like drones and killbots.
A major theme in cyberpunk is national boundaries get blurred and security threats come from independent groups rather than governments. Which is basically exactly what we've been dealing with for the past twenty years.
There's a good chance though the strongest military powers--corporation or government--would have large and varied militarizes for the purpose of fighting and winning any kind of war. Because "plan for everything" is generally a sound military strategy if you have the money for it.
Massive numbers of mercenaries and independent contractors die in a variety of new and interesting ways.
what happens when that setting expands throughout a whole solar system?
Independent contractors, terrorist bombings, false flag attacks.
I think you should check out Infinity. The game setting is basically exactly what you're looking for. Large scale warfare is less common, with most conflicts being a series of deniable ops, often done while the public conflict resolution is happening at a negotiation table.
Ghost in the Shell was mostly proxy wars in the parts of the world that don't do the whole cyborg thing.
So small squads, high degree of competency amongst its members, operating on at least a modicum of secrecy. Armament depending on the situation.