Backup and restore MMS
If your Android is not root, you can try another pc program called Backuptans Android SMS + MMS Transfer. Connected your phone to computer via usb or wifi and then click the backup button to save messages on computer. And yes, MMS restoring from computer to Android is supported. Tested with my Galaxy S3. 

PS: It also can work for an Android phone with broken screen. Just make sure your have enabled usb debugging on device first.