How migrate Builds with Artifacts to another Teamcity server.

Hello,

In order to do that, you would need to:
a) ensure the same build configurations are present on the other server;
b) copy over the build metadata;
c) copy over the artifact files.

The easiest solution for that might be a server copy; please see an article on how to make a copy of existing installation on a remote server here - https://www.jetbrains.com/help/teamcity/how-to.html#Create+a+Server+Copy. 

If you provide me with some details on what you would like to achieve, I will check if there are other approaches that could suit you more.