On October 22, 2018, New York State Department of Transportation [#NYSDOT], in a Facebook post, became yet another major body to publish a description of the ‘Move Over Law’ that can be seen as being too confusing.

They wrote: “Slow down and Move Over for roadside vehicles with flashing lights,” but this isn’t quite right, and it even drew questions from confused individuals on Facebook. Indeed, under some circumstances moving over and slowing down could potentially increase risk rather than the opposite, so use good judgement.

This extremely important rule always needs to written accurately, so that nobody can misunderstand.
As soon as you see an emergency or maintenance vehicle on the shoulder, up ahead, make a safe lane-change to the left so that you leave an empty lane between yourself and that vehicle. If the lane on your left is too busy for you to be able to do this, or you are on a two-way road, then slow down a lot before you reach the emergency vehicle. A common guideline, at highway speeds, is that if you have to stay in the lane closest to the emergency vehicle/s, drive at least 20mph slower than the posted speed limit.

It is a good idea to do this for all broken-down vehicles, too, because somebody may be working on them or might get out of the vehicle unsafely, on the side nearest you.
You need to be aware of the relevant state or national laws, too.

The important thing is to be observant, see problems early, and do your ‘move-over’ safely and in plenty of time.
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I can tell you that I debate this often with Police, driving instructors and graduates of my instructor courses here in my own jurisdiction. The priority is, as you say, to first move over safely and blend with the traffic in the adjacent left lane. This needs to be done as soon as the need ahead becomes apparent, smoothly by matching the speed, often faster, of traffic in the next lane as in any proper lane change. By driving with eyes high and scanning far ahead this is rarely a problem except for example on a blind curve or with some other visual obstruction. Then the slowing down becomes the priority but as gradually as safely possible.
Thanks for your welcome comment, Bill.
In the photograph which is published which I assume to be either an example or even of the location referred to, there is no Buffer Zone coned off from live traffic.
This shows that the US regulations regarding Temporary Traffic Management of Work Zones is not policed correctly or is itself lacking in safety.
Time to get it right, yanks!!
Just seen the second photo of the guy fixing a wheel of his truck at the side of the highway. Can I further comment? In the UK on our motorways, the advice now contained in the UK Highway Code in this situation is for the occupants of the vehicle to leave by the door(s) away from the traffic and wait on the verge or embankment behind a crash barrier if there is one and wait for a traffic office to arrive. Traffic officers are not police, they are “safety wardens” employed by Highways England to patrol the motorways and trunk roads to deal with such emergencies. The Traffic Officer will organise an appropriate service agency to come and fix the problem, even to change a flat tyre, so that no-one has to run the risk of being exposed so near to live traffic.
Lance, there’s certainly nothing like that nationally in the USA, nor do I know of any individual states that have such a scheme. There is no equivalent to non-police ‘traffic officers’ in the USA.