To copy an entire WordPress install with all of your content, plugins, settings, and everything else, you can use a plugin like All-in-One WP Migration.
Once you've created a .wpress package with All-in-One WP Migration, import the package on the target site and do the following:
Save your permalinks twice via Settings > Permalinks
Resign your shortcodes via Oxygen > Settings > Security
Regenerate your CSS cache via Oxygen > Settings > CSS Cache
Check to verify that everything looks & works as you expect
You may also need to open each template in the admin area and click the Update button on the right-hand side, but this is rare if you have followed the above steps.
If you just want to copy Oxygen settings or some page designs, use the methods below.
Regardless of the migration method used, you may need to re-sign your site's shortcodes for all elements to render properly once it is moved. See "Re-Signing Oxygen Shortcodes" below for instructions.
Copying Oxygen Settings
You may easily copy Oxygen's settings (global styles, classes, custom selectors, and style sheets) from one install to another. Follow these steps:
On the original install, select Oxygen > Export & Import in the WordPress admin panel.
Copy the text from the Export box.
On the new install, select Oxygen > Export & Import in the WordPress admin panel.
Paste the text into the Import box.
If you want to import Global Colors, tick the Import Global Colors checkbox below the Import box. This will overwrite all currently present Global Colors on the target site.
Click Submit.
Copying Individual Page & Template Designs
To move a Template, Page, Post, or any custom post type, browse to the Edit screen for that post, then click [+] Shortcodes.
Copy the shortcodes.
Then, create a new Template or Page on the new website, and paste the shortcodes into the same [+] Shortcodes box.
Note that if your design relies on any of Oxygen's settings, you must also copy those to the new site, as per Copying Oxygen Settings above.
Moving All Templates & Other Site Content At One Time
To move all content and templates from one site to another, you should use the built-in WordPress import and export tools.
First, on the original install, select Tools > Export in the WordPress admin panel and follow the process.
Now, on the new install, select Tools > Import and follow the process.
Note that it is usually necessary to copy your Oxygen settings to the new install, otherwise your page designs will be missing your global styles, classes, etc.
Re-Signing Oxygen Shortcodes
If parts of your design are missing after migrating a site, you will need to re-sign your Oxygen shortcodes.
Before proceeding, perform a full backup of your site.
Once your site has been backed up, go to Oxygen > Settings > Security in the WordPress admin panel and ensure that the "Check Oxygen's shortcodes for a valid signature before executing" checkbox is checked. If it's not, check it and then click Save Changes.
Next, click the "Sign All Shortcodes" link just below the Save Changes button.
Now, check the "I have made a complete backup of my site" box and the boxes for each post type on which you'd like to re-sign the shortcodes. Usually, you'll want to re-sign shortcodes on ct_template, page, and post.
Click the Start shortcodes signing process. You'll be able to watch the progress and see when the re-signing of Oxygen shortcodes has been completed.
Once the process has finished, check the posts or templates that were missing elements and those elements should be visible and editable again.