DTTG on iPad and iPhone are doing fine and reporting no issue respectively triangles.
DT on Macbook ist doing fine and reporting no issues.
DT on the Mac Mini has no messages in Log. But its changes are no shared with the other devices.
I had run repair and made integrity test on all installation. There were wrong checksums and a dozen files, which are sorted out.
So it dit look in the logs and cleared the problems, before I shared my issue.
Note:
Only the global inbox is affected! Items and group within the databases are doing fine.
This is currently my workaround to get items from my main workplace (mac mini) to the other installations.