The Backup section of iExplorer is designed to export copies of that data to your computer for archiving or sharing and is read-only which means you won't be able to add any of it back to a device via the Backup section. For transferring text messages, voicemail or notes to an iPhone or iPad and back into their respective iOS application, the only way to do that is to restore the device from an iTunes Backup that contains the data, however this will restore and replace all data. There's no way to selectively pick and choose which data is restored due to the way that Apple designed iOS and iTunes. If you would like to put all Backup data from one device onto another one and set it up the exact same, this Apple article explains how:
The only Backup data you can selectively transfer back to your iPhone or iPad and into their respective iOS application is contacts, calendars, and bookmarks since you can just import any of that data to a desktop application and then sync them to your iPhone.
There is some data that's not backed up in iTunes such as your iTunes media and synced photo albums that won't be restored on the new device unless your computer has a copy, so you'll still want to use iExplorer to transfer that data to your computer before restoring to a different iPhone, iPad or iPod. iTunes media includes music, movies, playlists, TV shows, podcasts, audiobooks, iBooks, music videos, etc.
TEXT MESSAGES, VOICEMAIL, AND NOTES
If you just need to view a history of the text messages on your device and don't need to continue the conversations, then you could use iExplorer to transfer the exported text message files to an iOS app. We have an entirely free iOS app called iExplorer Mobile that can read PDF, TXT, and CSV files and more:
The same goes for voicemail and notes. If you don’t need the voicemails in the Phone app or the notes in the Notes app, you could just add them to any iOS app that supports audio or text files such as our free iExplorer Mobile app. Another option for voicemails is to import them to iTunes and sync them like music tracks to your iPhone.
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