It's a bit surprising, really, that it's not easier to export and save entire iPhone text message conversations or at least large portions of a chat.
With a quick workaround, however, you can indeed save entire text conversations had on your iPhone, and you can then forward them as an iMessage or even email them to yourself.
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How to save an entire text conversation on your iPhone
1. Open the text chain you wish to preserve and hold a finger down on one of the texts in the conversation.
2. Tap the "More…" option when it appears, then tap the circle to the left of each text and image you want to save.
3. Tap on the curved arrow at the bottom right corner of the screen, then enter the phone number or email address that you wish to send the text the conversation to.
4. You can also hold a finger down on the new text message and tap "Copy" to copy it for pasting elsewhere on your iPhone, such as into an email or a note.
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