Old question, but for this use ACF Pro. This will allow you to easily export fields and import them to the new site. When you need to make changes you can just import the fields back into a dev site, or keep the dev site available for making changes. Then make those changes and do the export and import back to the client’s site.
Personally, if a client wants to host a site themself I export the entire database when moving it and import the entire thing so it’s all set up once I move the files, then you don’t need to worry about putting ACF field groups into PHP. However the JSON export and import in Pro is great for this type of thing. If you still want to load the fields using PHP you can do this by reading in the JSON files from the export and register the field groups from that.
Hope this helps someone.
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