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In most cases the issue is related to processing downloaded bases on the server drive.
Databases are downloaded from our sites successfully, but the problem appears during compiling and copying the downloaded bases locally on the KSE server.
Such behavior may be caused by the following:

1. Not configured exclusions for KSE in Kaspersky Security for Windows Server or Kaspersky Endpoint Security.
2. Other utilities (backup, for example), that may interfere with the file processing.
3. Incorrect operation of the delete function on high-speed drives, for example, SSD drives.

Solution:

1. Configure Kaspersky Security for Windows Server or Kaspersky Endpoint Security for correct simultaneous work with KSE. Completely exclude the KSE folder, all subfolders and KSE processes from the scan scope: 

Kavscmesrv.exe 
Antiphishing.OutprocScanner.exe
Antispam.OutprocScanner.exe
Antivirus.OutprocScanner.exe
Kse.Ksn.exe
Kse.Licensing.exe
Kse.Updater.exe 

2. Set the startup type of KSE service to manual.

3. Stop the KSE service.

4. If issue is related to corrupted AS bases, delete all contents from the folders:

C:\Program Files (x86)\Kaspersky Lab\Kaspersky Security for Microsoft Exchange Servers\data\bases\as\bases 
C:\Program Files (x86)\Kaspersky Lab\Kaspersky Security for Microsoft Exchange Servers\data\bases\ap\bases
C:\Program Files (x86)\Kaspersky Lab\Kaspersky Security for Microsoft Exchange Servers\data\bases\ap\cache

If issue is related to corrupted AV bases, delete all contents from the folders:

C:\Program Files (x86)\Kaspersky Lab\Kaspersky Security for Microsoft Exchange Servers\data\bases\av\bases
C:\Program Files (x86)\Kaspersky Lab\Kaspersky Security for Microsoft Exchange Servers\data\bases\av\cache

If issue is related to corrupted AS and AV bases, delete all contents from the all folders:

C:\Program Files (x86)\Kaspersky Lab\Kaspersky Security for Microsoft Exchange Servers\data\bases\as\bases 
C:\Program Files (x86)\Kaspersky Lab\Kaspersky Security for Microsoft Exchange Servers\data\bases\ap\bases
C:\Program Files (x86)\Kaspersky Lab\Kaspersky Security for Microsoft Exchange Servers\data\bases\ap\cache
C:\Program Files (x86)\Kaspersky Lab\Kaspersky Security for Microsoft Exchange Servers\data\bases\av\bases
C:\Program Files (x86)\Kaspersky Lab\Kaspersky Security for Microsoft Exchange Servers\data\bases\av\cache

5. Set the startup type of KSE service back to automatic/ automatic (Delayed Start), as it was before adjustment at step 2.

6. Start the KSE service.

7. Update anti-spam or/and anti-virus bases manually through KSE Management Console.

8. If the issue persists, contact Kaspersky Support.

And a little bit more:

C:\Program Files (x86)\Kaspersky Security for Microsoft Exchange Servers\data\bases\ap\backup\temp

C:\Program Files (x86)\Kaspersky Security for Microsoft Exchange Servers\data\bases\ap\Bases\temp

C:\Program Files (x86)\Kaspersky Security for Microsoft Exchange Servers\data\bases\ap\corrupted

 

C:\Program Files (x86)\Kaspersky Security for Microsoft Exchange Servers\data\bases\as\backup\temp

C:\Program Files (x86)\Kaspersky Security for Microsoft Exchange Servers\data\bases\as\Bases\temp

C:\Program Files (x86)\Kaspersky Security for Microsoft Exchange Servers\data\bases\as\corrupted