Add a small groundTexture, that should make them moveable.

I'm late, but I still don't know how to do that.
I was on hiatus for a few months and now I want to fix this, but no avail yet.

You don't need it anymore, Construction Mover does it for you now.
If you want to do it by hand, add
groundFaces = {
{
face = {
{0.1, -0.1, 0.0, 1.0},
{0.1, 0.1, 0.0, 1.0},
{-0.1, 0.1, 0.0, 1.0},
{-0.1, -0.1, 0.0, 1.0},
},
modes = {
{
type = 'FILL',
key = 'shared/asphalt_01.gtex.lua' --'shared/gravel_03.gtex.lua'
}
}
},
}
to the data, which updateFn() puts out.

Still no avail.

The cons already got
which I changed to this

Strange. If there are already groundFaces, you don't need to add more. Perhaps that mod plays some tricks with game.getInterface()?

Nothing with getInterface found in any of the files, incl. the mod addon and the assets pack (just to be sure):
https://gtm.steamproxy.vip/sharedfiles/filedetails/?id=2128272632
https://gtm.steamproxy.vip/sharedfiles/filedetails/?id=2797825680
https://gtm.steamproxy.vip/sharedfiles/filedetails/?id=2685102191

I checked it, those are bridges and bridges cannot be moved. Pillars and buttresses probably can because they are normal constructions, but just those two.

It's only about the buttress and pillar assets. Those I want to move and cannot.

I can, it must be something little and stupid that escaped you. I did not change the mod, I only used the Con Mover with the new parameter on,

Darn. Some mod is indeed wrecking my game, I also have other issues:
Some industry mods with integrated stations - like Rail & Track industries or the 5-in-1 industries from hugedragonyk - have no "configure" button anymore when in the map editor and no "station icon" anymore when in game.
Perhaps those issues are related somehow.

The problem is that the stdout.txt does not provide anything useful:
https://www.dropbox.com/scl/fi/mp9546vfexcvo0dw0yorj/stdout-2024050301.zip?rlkey=bhuj1qn3sfuea6hmo0z0oui7w&dl=1

If I snap attach any of those at a railroad track it works with no issues, fyi.
The issue only occurs when I snap attach to a road.