Hey there wunwspirit,
Since you're unable to download messages from another device, like a computer, you would need to have a screenshot of the messages or forward the individual messages to another account or device via email. The steps below will show you how:
Forward messages
1. Touch and hold a message bubble you want to forward, then tap More.
2. Tap to select one or more messages to forward, then tap 
.
3. Enter recipients, then tap 
.
Send screenshots of conversations
To show the message bubbles and timestamps just as they appear in a conversation, send a screenshot as an attached image instead of forwarding messages.
1. Scroll to position the conversation on the screen, then take a screenshot.
2. For longer conversations, scroll again to reposition the screen and continue taking screenshots.
3. Tap 
, then tap 
.
4. Enter recipients, tap the text field, then tap 
.
5. Tap images to select screenshots, then tap .
To share your messages on other devices besides your iPhone, use Messages in iCloud.
Forward and share messages on iPhone - Apple Support
You're always encouraged by Apple to share feedback for future consideration as well. Review the policy for feedback and submit it using the guidance here:
Legal - Unsolicited Idea Submission Policy - Apple
Product Feedback - Apple
Take care!
