First thing off,
That feature is not retroactive. This means that you have to set global or local from the onset, knowing in advance what you want to happen. Personally, I like to have Global by default when I write. I tend to use local when editing parts in engrave mode, but that’s me. If one makes an error, one has to turn off the override, change the Global/Local setting, and then apply the override.
When you have global set, and you are writing, any changes you apply away from a default (for example, you force a dotted slur on a single slur through the properties panel) will appear in the corresponding layout (if it applies to individual staves) or layouts (if you have executed the change on a system object).
If you have local set before altering that slur, once altered to a dotted slur, the dots will appear in the score only, and the slur will be undotted in the part. Generally speaking, this is more useful when editing a part; meaning that you would change a properties on a part while “Local” is set, and the properties you change on that part will not appear in the score. Even if that slur is linked to another staff, only that layout will have that alteration. Same with system objects. In other words, for a system text to appear in the score but not the parts, you would have to set Local, write the text (shift-alt-x) on the score, press escape, select it and set it to “Hide” with the properties panel. To do the opposite, go to a part, set to Global, write the text, go to the score, set to local and flip the hide switch.
This has nothing to do with flows, but rather with elements that are on the score, where they appear. It is especially useful when modifying slurs in engrave mode so the alterations only appear in the parts. Or in write mode, having playing techniques hidden in the score (if independent voices playback is on) but appear in the parts. Or having pizz. al fine written with a PT suffix in the part, but not in the score etc …
Note that some properties panel overrides will always remain global, such as dynamics suffixes. The way to tell local settings will not apply is that, when in Local mode, the text beside the panel’s switch will remains bold white. When you see that, it means that this in an instance where having independent properties between score and parts is not allowed. properties that are changed globally will always have that label in bold, while local properties will be in thinner white.