Most of these levels will have officers receiving orders from above, interpreting them, and issuing orders to lower echelons. But these events won't all happen at the same time.
To take the D-Day landings as an example, the supreme commander (Eisenhower) and the initial Army Group commander (Montgomery) will have made their main plans, in consultation with each other and their respective staffs, months in advance. The Army commanders, Bradley and Dempsey, will have made their plans on that basis, still well in advance, and issued orders to their corps commanders. There was one corps per beach, so each corps commander is fairly independent of the others.
For a beach, the corps commander will have allocated his divisions in consultation with his staff and the divisional commanders and staffs. That gets us to the divisional level.
Then we hit a difference in terminology and organisation. The Commonwealth divisions were made up of brigades, each of which contained several battalions. The American divisions were made up of regiments, each of which contains several battalions.
A British-tradition regiment is an organisation that creates and trains battalions, which are then assigned to brigades, but almost always to different brigades. It isn't a combat unit as such, but a centre for training and traditions. A US regiment is a combat unit, and has the same set of battalions, almost all the time.
In any case, the divisional commanders would give objectives to the commanders of their brigades and regiments, and those commanders would allocate the objectives to battalions, probably holding a battalion in reserve. We're still a couple of months in advance of the invasion here.
The battalions are the tactical units. Their commanders would do the detailed planning for the attacks in their areas, and would hold map exercises to test their plans. The troops, meanwhile, would be practicing disembarking from landing craft, and rehearsing their attacks on dummy versions of the targets. These exercises will inevitably find problems with the plans, which will have to be adjusted.
On the day, the company, platoon and squad levels of command are on the ground with their men. Their jobs are to take their targets, finding other ways to do the job if the defenders resist, helping out neighbouring units if they can, and calling for support if they need it. The battalion HQs may be on the beach or on ships, and their job is to decide where to commit their reserves, and to tell higher HQs if they are having problems.
Brigade/regiment, division and corps HQs are on board ships. They might have two levels on the same ship, but they definitely won't have all three, because that means too much is lost if a ship is disabled or sunk. Their job is to feed their own reserves to lower levels, and tell higher levels what's going on. Army, army group and supreme command are in England. They need to stay out of the lower levels' hair and let them get on with the fighting.
Once Eisenhower had given the "go" order, he didn't need to give any more unless there was a disaster: everything had been pre-planned and everyone knew what to do. The orders that it would have been meaningful for him to give would have been endorsing the army group's decision to abandon a beach that could not be taken, and abandoning the whole attack, which is the supreme commander's decision. Giving orders to commit a reserve battalion or company is not his job - he doesn't have as good information as the commanders closer to the action, and he needs to keep track of the overall situation.
Supreme command involves a lot of responsibility without the ability to do very much to help on the day.