There are several ways to do it.

I have public IP address (dynamic) from my ISP and one old laptop as server. This server run mysql database.
Then I have a shack computer that runs also mysql database and cqrlog.

The shack computer has main cqrlog database (unchecked "save log data local machine"). It saves it there anyway, but using mysql port 3306.

Server mysql and shack mysql do have 2 way repilcation with databases. So changes can happen to other database and it will replicate it to another.
Even afterwards if another computer is shut down. That is usually shack computer. Serever is running 24/7.

So I can connect to my server via internet in 3 ways. Https, shh and openvpn.
At servers end http service is running cqrweblog https://www.dl8bh.de/cqrweblog/
By using that I can log qsos from any computer that is connected to internet. Even with my smart phone! Qsos go to servers mysql database and are replicated to shack computer when I return and turn it on.

I can also run cqrlog on some other computer that is connected to internet. In that case I open either ssh tunnel to my servers mysql service with command:
ssh -L 3306:localhost:33060 mydyndnsname.server.com
and adjust cqrlog to open database from localhost port 33060 that points to my server via ssh tunnel.
After that I can log qsos and when I return and turn on shack computer its mysql server will replicate with my servers mysql server and get my remote qsos.

Third way is to use openvpn tunnel between internet computer and server at home. In that case no changes are needed in cqrlog settings it will run with same settings as at home because openvpn tunnel extends local LAN via the internet. And again this remote internet computer logs to server database adn when I get home and turn on shack computer its mysql server will replicate with my servers mysql server and get my remote qsos.

I have used all ways.
ssh tunnel is faster than openvpn over slow mobile networks.
Web browser logging via cqrlweblog works well even over 2G mobile networks.

Following knowledge is needed and can be googled very easy:

- using dynamic dns with ISP connection
- setting up apache web server
- setting up cqrweblog
- mysql database replication between 2 mysql servers (in both directions)
- ssh tunneling
- openvpn setup and usage

In your case easy start is to set base desktop run cqrlog databases from mysql servers port 3306 using "real" mysql connection instead of "saving log data to local machine" that uses separate mysql pipe to access same database.
You need to export your logs, then import them again when using port 3306 mysql server.
See this, it tells how to start without "saving log data to local machine": https://www.cqrlog.com/comment/6776#comment-6776

When it runs ok, you will probably have also running apache webserver in your desktop coputer. If not install it from package.
When it is installed downlaod and install cqrweblog.
When it runs OK via local LAN you need public internet address (dynamic is ok).

Using dynamic dns (several services available) you give name to your dynamic ipaddress.

When that is ok and you can resolve your IP address based the name you have you have to open your router's NAT so that cqrweblog request from internet goes to your desktop.

I recommend NOT to use normal http or https ports (80 or 433), use port number over 10000 (that you can remember) and username/password autenthication at cqrweblog http page.

Afteer all work you can youe your rasperry pi, or your smatphone to log qsos. Desktop must be running while you are away.

These are very brief instructions, but you can resolve all step by step with the help of google.
That is the way I once did it by myself.