If the "assets"/project entities reside in a "project", AND the project is relatively small < 2Gb repos+DB, you can "export" your project from Site Admin into a "QCP file.
Then, you can "create a new project" in the other server from the QCP file.  Then can copy "assets" between projects (both on the SAME server) as usual.

Starting with QC12.6, that version can read in 12.xx generated QCP's to create projects in the case your two servers are different versions, but the TARGET server must be 12.6 or later. 

IF project is very large >>2Gb, you can make a backup of the project DB + zip of repos folder, then RESTORE to other ALM server.
Then can copy "assets" between projects as usual.  this is similar to what you do to migrate projects to a new server during an upgrade.

Generally, you can perform copy/paste of project entities across TWO instances of Internet Explorer pointing to two DIFFERENT projects, BUT it usually does not work when the projects are in tow different ALM servers.  -- you could try it first, I suppose.
Remember, do NOT do "drag-drop", it must be rt-click-copy on entity(s), then rt-click-paste into other project.