Rounding happens when shift is not held, keeping the keys on integer frames. This means not all the keys move the same number of frames. The handles get scaled slightly different when that happens. The Curves view shows a quite distorted view. When one key moves more than its neighbor, the curve changes slightly. This is consistent with other curve manipulation.

Note that your curve is very simple, since the handles are flat. How the handles are adjusted has to work in all scenarios. Be sure to also try a more complex scenario, like:
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When you hold shift there is no rounding and your keys can be on fractional frames. If you click a moved key and look at the Current frame text box in the dopesheet, you can see the keys are not just moved to 0.1 but eg 25.6974. When all the keys move the same amount, the handles are scaled evenly and the Curves view doesn't change.

3.8 works differently. The handles are stored differently and it doesn't support intelligently adjusting the handles. It's not just a matter of doing the same thing in 4+.

If you look at the handles in the Graph view, you can more easily see how they are being manipulated. The graph is the actual value over time and that is what matters for the movement in your animations. What the Curves view shows is the rate of change over time. That is useful, but deceiving in many ways, making it less important. The difference in the graph view when key frames are rounded is negligible and the logic looks correct, so I don't think we have a bug to fix here.

4.2 has a lot more improvements versus 3.8 than just physics!

By "flat" I mean this button:
http://ko.esotericsoftware.com/spine-graph#%ED%94%8C%EB%9E%AB
It looks similar to the "separate handles" button, but it's not the same. Flat is when the handles are both at the same Y position in the graph.

We would say handles can be either "straight" (the default) or "separated". I know we are speaking different languages, but I just wanted to cover this briefly so we're sure we're talking about the same things.

When the handles are flat, the scaling is simpler. Some aspects of scaling are only seen when the handles are not flat, regardless of whether then handles are separated.

Even if the spacing between frames is adjusted, I want the shape of the curve to remain the same.
This is probably what the animators want.

Actually that is not usually the case. The curves define the rate at which the keyed value changes. It's most important that the rate not change abruptly. To do that you need the handles to be straight through the key. Most of the time that is more important than keeping the relative shape of the curve as seen in the Curves view. That is why the keys are not separated by default.

Again I encourage you to use the graph more to visualize your curves. See how your curves flow throughout your entire animation is extremely useful. It's great that you are paying so much attention to the Curves view, especially when you find bugs that we can fix, but I strongly believe that the Graph view has a bigger impact on your animations. Not only that, but it allows you to better see what is happening when you move keys around -- you might find bugs more easily!

through the process of narrowing and widening the frame gap

If you scale your keys with rounding, those are the new key positions. If the keys were not moved an equal amount due to rounding and you later try to scale them back to where they were, it is likely they will be off by a frame. Scaling the keys with rounding changes the spacing between the keys, unless the scaling is evenly divisible (50%, 200%, etc).

That is expected behavior and how it works in 3.8, except I see that in 3.8 letting go of the mouse button and then scaling again remembers the original positions. If you dismiss the box selection, and make a new box selection, then you'll see the same behavior in 4+.

In 4+, after letting go of the mouse button, moving the same box selection does not use the original key positions for scaling, so you see the keys being off by 1 frame. We should be able to improve this!

Thanks for your input! It's always nice to improve things.

3.8 is absolutely ancient now. Just go fully to 4.2! Physics! What if you could have folders for constraints?! Maybe draw order too!