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You can forward them to email. But it's easier to just copy and paste.
This issue is proving to be my biggest regret switching to iPhone from Android. On the Android phone I used an app that forwarded all my SMS messages, incoming and outgoing to email. Similarly, the app forwarded details of each call. This meant that I had a simple permanent, automatically backed up record of all my SMS messages and calls. The app was free. It is called SMS backup plus.
I cannot find any similar app for the iPhone. There are apps that will export the iMessage files as Excel spreadsheets, but that is not what I want. I want the phone to send an email for every single iMessage. I wonder if Apple is deliberately obstructing this?