What you can actually do is exporting the relevant text frame to ICML (InCopy markup). ICML files can be placed multiple times⁰, and when you try to edit their content, you'll end up having to check out the frame⁺ and afterwards checking it in again. Any other placement of that file will then show up as an outdated link that needs refreshing. You can also use InCopy to edit the ICML file alone, by the way. More information can be found at https://helpx.adobe.com/indesign/using/sharing-content.html.
If you try to place an ICML file within a text frame, though, you'll end up with an unlinked copy, which means synchronisation won't work. As a workaround:
- create a temporary InDesign document
- place the ICML file directly as a linked text frame
- mark the frame and export that as IDMS (InDesign snippet)
- no need to save the temporary InDesign document
- place the IDMS snippet inside any text frame you want
The IDMS simply contains the information "text frame linked to ICML file", and that can be placed inside another text frame to maintain the linked property and thus the synchronization. Just don't try nesting linked text frames within linked text frames, that seems to horribly fail...
⁰ You should however not do this multiple times within one document, otherwise you'll end up with a weird re-synchronisation feedback loop...
⁺ Which effectively means locking the ICML file so you or your colleagues don't accidentally modify the same file synchronously which would lead to content conflicts.