Texts and photos in text messages do not automatically sync with Google or transfer to a new device. A photo from a text will need to be saved from the phone to sync with Google or transfer. Texts can be backed up and transferred using third-party apps available in the Play Store.
To save photos in texts, see How to Save a Photo Sent to You via Text on a Motorola 2.0 Phones
To backup photos saved from texts, see How to Back up Additional Folders to Google Photos
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