Is that if the service does it, or if the users do it?
When I moved my photos from Flickr to SmugMug a few years back they (SmugMug) had a browser plugin that moved everything over. I thought it was odd, but figured it was to get around some kind of rule like that.
Yes this could work, and even if the API terms do not allow it (ie Google Photos API do not allow you to download the pictures for storage) you can use the Data Export feature that most websites have and use this export to upload back your data elsewhere.
