During a conflict, the military is the brunt arm of the political will. As the conflict ends, the role of the military must drop much more quickly.
Logistics, Supply, Military Police, Chow, HQ & Support, and possibly Construction and/or Transport are left behind. Not combat units, they begin collecting in central locations for transport home as soon as possible.
Who should run perceivably-occupation-like peace operations? It should be, basically our national Police Corps, like a Gendarmerie (Spain's Civil Guard especially) of Europe. In other words, Peace Officers, to keep the peace the military has so recently won. We don't have this today, and multi-lingualism might be a requirement.
During the initial days of a transition to Police Corps, of course, there will likely be a shortage of peace officers, and military police should be used.
I'm not sure fitness, of their units, is a judgment we should be making. Maybe offer our police-training, rub by our military, for any units that do not do well.
State, if things stay stable or improve, then takes an ever increasing role. If one does not invade the wrong targets, in the first place, peace operations are greatly simplified.
One of State's early tasks is to clear the rubble. Local equipment should be used first. Equipment totals, per owner, should be simultaneously reviewed, including drive test, before starting. Military Construction Battalions might be able to lend a hand.