Is there a way to export syntax color settings from one Xcode installation to a new one?

As of Xcode Version 11.6 (11E708), the file extension for color themes is xccolortheme.

The path to the included themes is:

/Applications/Xcode.app/Contents/SharedFrameworks/DVTUserInterfaceKit.framework/Versions/A/Resources/FontAndColorThemes

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The path to a user's custom themes is the same as in the accepted answer:

~/Library/Developer/Xcode/UserData/FontAndColorThemes/

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To install a custom theme, you can simply drag the xccolortheme file into the folder for custom themes; quit Xcode, then restart Xcode. The theme will now appear in Xcode's Preferences for Fonts & Colors.

With Themeinstaller, you can just open .dvtcolortheme files and they will be installed in your Xcode folder.

In case you wanted some new themes there are many to be found on codethemes.net, having Themeinstaller, you can install them, just by clicking "install" under any of your choice.

I found this answer while looking how to move the new (at the time of writing this answer) Civic theme in Xcode 8 to Xcode 7.3. Now themes are bundled inside of the Xcode application's bundle at following path:

Xcode.app/Contents/SharedFrameworks/DVTKit.framework/Versions/A/Resources/FontAndColorThemes

Also Xcode8 theme has extension xccolortheme while Xcode7.3 has dvtcolortheme. From my experience - it's safe change the extension without modifying the file itself.