Here is detail behaviors after improving.

User should config notifications group to receive exception msg in Notifications app. 

Before behaviors:

• The sync will stop immediately when found one renamed/moved node (nodes has been renamed on author and is not present on public)
• No meaning full log error. See 

  ERROR info.magnolia.module.scheduler.CommandJob : Cannot execute command [synchronize] for job [Synchronization] in catalog [synchronization]
  

• Users in notifications group will receive one notification msg as below:

After improving:

• Don't stop sync. Continue to sync for normal nodes. After finishing sync, 1 log warning will show all the list of moved nodes that can't sync. 

  info.magnolia.synchronization.commands.SynchronizationCommand 31.05.2021 11:06:56 – The following paths [/node1/node3, /node1/node10] have been moved from the last published original path. Please, publish manually before synchronising.
  

• Users in notifications group will receive one notification msg as below:

NOTE: Update both 6.2 and 5.7 docs.