How can I synchronize entries between servers?

DAVx⁵ is a client that synchronizes data (contacts, events, tasks) between a CalDAV/CardDAV server and the local Android storage.

So, it's not in the scope of DAVx⁵ to synchronize data between different servers. However, there's alternative software for that use case:

pimsync

There's pimsync (formerly vdirsyncer), a command-line tool that can be configured to synchronize various storages and protocols, including local storage and CalDAV/CardDAV.

You can use it on device to synchronize between two storages, for instance between the local storage of the server and another server, or between two other servers.

It can be run periodically and also has a daemon function.