I happened to look at UA1175 NTSB report - 777 which lost an engine near HNL - and they did some seemingly chaotic things with throttles.
Image below is a snip from NTB data docket, red and green are throttle lever position, and graph covers ~30 seconds after engine failure. It is also said that the crew didn't have indications about which engine failed during that time other than aerodynamics and controls behavior.
What can be the logic of such back and forth movement and eventual settling to idle until things became clear? Is settling to idle a part of SOP or this was about "we have enough altitude, lets just glide until we understand what is going on"?
Source: https://data.ntsb.gov/Docket?ProjectID=96738 , item 11 FDR factual report, pages 11-12.

Any engine shutdown procedure begins with identify and confirm the engine to be shutdown. It’s not “chaotic”; it’s about moving the throttle to try to feel or see changes that would identify and confirm what they’re thinking is wrong and with which engine. You look for changes in vibration feel, aerodynamic reactions to thrust changes, if any; and changes in sound. Once they were agreed in the affected, confirm the throttle in on idle position, shut down the engine.

I would suggest that initially the thrust lever increase could be the auto-thrust increasing to compensate for the speed decay. Subsequent manual intervention, then lower setting / idle for the drift down or descent. We're talking a period of 20 seconds here, from an unexpected event. As said above, identifying the issue then dealing with it appropriately. Sometimes taking a split second immediate action results in people doing the wrong thing. It is often better to take some time to correctly identify the issue, so you can have appropriate solutions.

Speaking as a private pilot, that experienced the loss of the right main on the day of my solo.....
1. Aviate. Fly the airplane. Fly it safely. Altitude and airspeed will always be your friend. Remain calm and work the relevant checklist, if one, for the situation. No checklist, go to your basic airmanship and ensure the public's and your safety.
2. Navigate. determine your best course of action. In my case, we (my instructor was still with me) looked at a ditching near Homestead Bayfront Park, calling the Air Force Base to foam the runway, or which farmer's field was freshly plowed for the softest touchdown.
3. Communicate. Once you have achieved the first two, successfully, then make your radio call. Let the ATC, and other aircraft know of your intentions.

These have to be done in this order and calmly. Panic or do these out of order and your are endangering lives.

Strebav8or wrote:

Speaking as a private pilot, that experienced the loss of the right main on the day of my solo.....
1. Aviate. Fly the airplane. Fly it safely. Altitude and airspeed will always be your friend. Remain calm and work the relevant checklist, if one, for the situation. No checklist, go to your basic airmanship and ensure the public's and your safety.
2. Navigate. determine your best course of action. In my case, we (my instructor was still with me) looked at a ditching near Homestead Bayfront Park, calling the Air Force Base to foam the runway, or which farmer's field was freshly plowed for the softest touchdown.
3. Communicate. Once you have achieved the first two, successfully, then make your radio call. Let the ATC, and other aircraft know of your intentions.

These have to be done in this order and calmly. Panic or do these out of order and your are endangering lives.

When you say "loss of the right main" I assume you mean the right main wheel. If that's the case, why would you consider landing off airport or ditching as an option? That doesn't make much sense to me at all.

About the OP's post, GalaxyFlyer is right on. Before every engine shutdown you verify that it's the correct engine by retarding the throttle first. TransAsia 235 is a prime example of why you do this.

rendezvous wrote:

I would suggest that initially the thrust lever increase could be the auto-thrust increasing to compensate for the speed decay. Subsequent manual intervention, then lower setting / idle for the drift down or descent. We're talking a period of 20 seconds here, from an unexpected event. As said above, identifying the issue then dealing with it appropriately. Sometimes taking a split second immediate action results in people doing the wrong thing. It is often better to take some time to correctly identify the issue, so you can have appropriate solutions.

"There is no problem so bad that you can't make it worse."

Chris Hadfield

Or, as an old instructor engineer used to tell, if you don’t the problem just wait til it gets bad enough to be a recognizable problem and do that procedure.

GalaxyFlyer wrote:

Or, as an old instructor engineer used to tell, if you don’t the problem just wait til it gets bad enough to be a recognizable problem and do that procedure.

That's a good one.

The process of problem recognition is in many ways more crucial and difficult than problem resolution. If you know what the problem is, you typically have the tools to solve it. If you recognise the wrong problem, or even a non-existing problem, however, you can get into a heap of trouble.

SoCalPilot wrote:

Strebav8or wrote:

Speaking as a private pilot, that experienced the loss of the right main on the day of my solo.....
1. Aviate. Fly the airplane. Fly it safely. Altitude and airspeed will always be your friend. Remain calm and work the relevant checklist, if one, for the situation. No checklist, go to your basic airmanship and ensure the public's and your safety.
2. Navigate. determine your best course of action. In my case, we (my instructor was still with me) looked at a ditching near Homestead Bayfront Park, calling the Air Force Base to foam the runway, or which farmer's field was freshly plowed for the softest touchdown.
3. Communicate. Once you have achieved the first two, successfully, then make your radio call. Let the ATC, and other aircraft know of your intentions.

These have to be done in this order and calmly. Panic or do these out of order and your are endangering lives.

When you say "loss of the right main" I assume you mean the right main wheel. If that's the case, why would you consider landing off airport or ditching as an option? That doesn't make much sense to me at all.

About the OP's post, GalaxyFlyer is right on. Before every engine shutdown you verify that it's the correct engine by retarding the throttle first. TransAsia 235 is a prime example of why you do this.

It was on a Piper Cherokee Warrior. We were left with a substantial chuck of steel that is directly below the fuel tank......of course, with 6 hours worth of 100LL on board.
We were looking at reducing our risk of fire.