Hey Spencer,
In my opinion the plugin for moving the changes is not a good idea. What type of customization you have in the parent theme? Are there code level changes (template modifications)? Or the customization is within the WordPress itself? If you have not modified any parent theme template then you only need to install and activate the child theme and all of your settings will be there.
If you have modified the parent theme files then only identify those files and move them (do not delete the files from the parent theme) in the child theme in the same directories as they are in the child theme and that’s it.
For example if you’ve modified the _content-none.php file in the parent theme which is located in x/framework/views/global/
now to move it in the child theme, copy this file first then go to your child theme and follow the same hierarchy of the folders i.e x-child/framework/views/global/
and paste the file in there. If you don’t find any folder inside the child theme and create it to have the same location.
Hope this helps!