Does general military authority extend across different branches of service? How should you approach the offender?

Since UCMJ extends across branches of services, general military authority does as well. For instance, disrespect toward an Officer, Warrant Officer, or NCO is punished under the same respective articles of UCMJ regardless of the branches of the personnel involved. So in formally correcting such incidents, shy of UCMJ, general military authority applies.

For the person doing the correction, I think it's important for them to differentiate between branch-specific policy and broad military policy. If I remember correctly, at some point in the 2000's, it was acceptable for Marines to have their hands in their pockets while in uniform, but Soldiers could not. So a correction wouldn't make sense. So a Soldier trying to correct a Marine on that issue wouldn't make any sense (I think the USMC policy of hands-in-pockets changed to a ban). Now take a situation where a servicemember of any particular branch is acting a fool in public (harassing civilians in a store), while still wearing their uniform; not that harassment while out of uniform is acceptable either. Now they are violating a general military policy on professional behavior of which there should be no confusion over our responsibility to keep that person from embarrassing the services. I've had to correct the last example before. . .

Yes. But you better be absolutely, positively, 100% certain that whatever the ‘offender’ from another branch is doing is actually inappropriate before you start pissing on their cereal.

I don't know about the legalities regarding the UCMJ, but I jumped a bunch of young Marines waiting for a flight at Minneapolis IAP. They were young, dumb, and fuil of beer acting like idiots in uniform. Went up to them, identified myself as a TSgt at the time, and tightened them up.