Contacts/CardDAV Push Notifications and macOS Sierra Server strange behavior

I have Sierra macOS Server 5.2, and the same "user01" account configured in Contacts.app on two Sierra 10.12 macOS computers and a single iOS 10.0.2 device (iPad). My expectation is that change on one device (e.g. add new contact) is in couple of seconds reflected on all other devices configured with the same account. However, this is not always true.

Manual synchronization invocation (quit Contacts.app, start it again) works always correctly. However, changes on one device (e.g. add new contact) are not pushed to all other devices. Usually, the change is pushed and data are synced immediately only to last device configured with the account. Other devices configured with the same account are not updated automatically.

There are valid push certificates (push is working on all devices, it just seem to depend on which is configured sooner and which later).

The same applies to Calendar.app, calendaring events using CalDAV protocol, calendar push notifications and macOS server.

In the apsd process log, on macOS, it can be even seen that push message arrived to the apsd process, it was passed to Contacts process. The difference is, however, that AddressBookSourceSync process is not downloading any changes from the macOS Calendar and Contacts server. Sometimes, push message does not seem to arrive from APNS server (based on apsd log apsctl tool).