How do I move installed games to an external hard drive via the EA App (PC).

You have to trick EA App into verifying the moved game install without needing to re-download it:

1) Completely exit EA App from menu or system tray.
(Not from the [X] in the window's upper right corner; that only minimizes to tray.)

2) Move the game install folder to the new location, e.g.
C:\Program Files\EA Games\STAR WARS Battlefront II
to
D:\Games\EA Games\STAR WARS Battlefront II

3) Start EA App. It won't see the game install anymore.

4) Go to install the game. When prompted for the location, point it to the folder where you just moved the game files.  It will automatically append the game folder, so just point it to "D:\Games\EA Games" in the above example.

5) Instead of re-downloading the game, EA App will check the existing files in the target location

Install UI will read "Preparing xxxGB / xxxGB" (instead of "Downloading").

Since all games files are present and their checksums match the current version of each file, once the files are verified it will finalize the configuration and report "Installation complete".