Answer

Virtually all Core connection problems come down to one of four possible causes:
 

Firewall

Each new major or minor release version of Designer (such as 9.0, 8.4, 8.3, 8.2 and so on) is a complete install on its own - therefore, firewalls require rules for each. If your machine has a third-party firewall, make sure its configured to allow Designer v9.0. If you are using Windows Firewall, you'll get a pop-up when you first run a new version after installation, in which case you must check all options before continuing. You can either find and delete the inbound rules for Designer (in which case you'll get the pop-up again when you next run the application) or you can edit the inbound rules to make sure they allow Designer full access to the network in all conditions.
 

IP Address

For your PC to communicate with the Core, it must be on the same local network and must have an IP address that in the same range/subnet as the Core. You can check the Core's IP from Configurator or from the Core's front panel.
 

Network

It is possible that the network is configured to filter certain packets Designer uses to communicate with the Core. Try connecting your PC directly to one of the LAN ports on the Core.
 

Core Name

The name of the Core in the design must match the name of the Core on the network for Designer to push a file to it. You can either change the name in the design to match the Core's name or use Core Manager to change the name of the Core to match the name in the design.

Further Investigations

To isolate the issue further, please see Finding a missing Core.