Hi!
When I modify the horizontal position of single notes in the score, the displacement doesn’t appear in the part. Is there a way that the new position of the notes is also applied to the single part of the instrument?
I don’t think so. This is because notes spacing adjustments are done in engrave mode, and you are engraving a specific layout, which can have very different layout options and formatting then the other layout. The propagate properties doesn’t work, because it seems that those are not properties, but local adjustments. (I hope that some of my words make sense and are correct in some way… 
)
EDIT: see my post below that shows how to propagate those Properties (Voice column X offset)
Previous suggested method, which is not necessary (click to expand)
But you could use Note Spacing Changes instead , and it is possible to copy them to another layout (in Write mode).
Here an example:
Dorico file:
note spacing changes for fanned beams.dorico (1.3 MB)
OK! Nice solution!
But it affects all staves I guess. If I’d like to create a texture with a canon-like structure of those gesture it would be impossible?
Opps, you are right!
Hopefully someone has an idea on how to “propagate” the note spacing adjustments (or this is possibly a feature request candidate) Edit: see the post below
Or it would be nice if the fanned beams would also influence note spacing automatically (another feature request candidate?)
For now I think the manual annoying adjustments are unavoidable… Edit: see the post below
(or maybe you can create a macro, if all the groups are the same…? I will try…)
@Hannes_Kerschbaumer
With macro recording it works! So you need to make the adjustments only once during the macro recording (use TAB key to switch to single note editing and arrow key to move to next column etc… during the macro recording, and select the first square before starting the recording 
):
(example of applying the recorded macro)
Hi @Hannes_Kerschbaumer, I was totally wrong about propagate! Those adjustments are indeed Properties, and in particular they are Voice column X offsets, which you can propagate.
Those notes position adjustments that you do in the Note Spacing tool, are also visible in the Properties Panel in the Note Spacing section, and you can propagate them to other layouts, selecting the noteheads in Engrave mode.
Video follows…
I would think that one issue here is these are absolute offsets from a calculated position. Since the calculated positions/spacing are likely to be different in score and part layouts, an absolute offset which works in one might not be right for the other.
That is, let’s say that the noteheads happen to be 1 space apart in one layout, and you nudge them 1/4 space closer. In the other layout, because of the material in the system, the noteheads are only 3/4 space apart to start with, and making them another 1/4 space closer is too much.
Yes, and your explanation was very clear. I noticed this (and you can see the slightly different result in my last video, since the part note distribution is larger than the score). To have a similar result between score and parts, one would possibly need to have the same casting-off and same global Note Spacing (the global one in Layout Options) . But as you said, with possible multi-bar rests and other differences, it may be hard to obtain the same proportional distribution as in the score.