1st step: clear yourself about how you want to retain the existing content.
For example: a contact page on website1.com might be having /contact/ permalink and on website2.com the permalink could be same. So, when you migrate /contact/ from website2 to website1, what do you want WP to do if not overwrite then, it changes/appends the new content URLs to /contact-2/ etc. But in this case, your existing internal links will break and you will need to update the links everywhere.
Either start clean by downloading/exporting all the data (files + sql) if you just need a backup. OR get ready for permalink update hassle since you want both versions so, you’ll have /contact/ and /new-contact/ like pages.
2nd step: considering you’re clear how to retain and know the consequences regarding permalink changes.
- Install new design dependencies (plugins/theme or tools that you used for a new design) on website1.com such as Elementor and probably its related extensions.
- Configure the dependencies i-e: Set the Elementor/others settings as you set on website2.com to make sure that when you import newly designed pages, they get the same effects.
- Export pages, posts, or other designed pages from website2.com using WordPress Export option available at WP Dashboard > Settings > Export
- Import them on website1.com using WP Dashboard > Settings > Import
- Flush permalinks for once from Settings > Permalinks > Click ‘Save Permalinks’
- Verify the new style/design or new page from the frontend.