That is not possible with Toolset.
Toolset lets you create, display, and even export the structure of those data sets.
However, the data is stored by WordPress, in native WordPress (or custom) Database tables, and belongs to the content, not the structure, which is what Toolset lets you export/import.
In that sense, Posts content (participants) should be exported with the Tools > Export of WordPress For example, but they do not allow to filter by our relationships or any other relations.
The particular scenario where you want a button or other mechanisms to export a certain set of posts only, needs custom code.
Additionally you may be interested in this:
https://toolset.com/documentation/user-guides/how-to-import-content-into-wordpress-using-csv/how-to-import-data-from-a-csv-file-using-the-wp-all-import-toolset-types-add-on-plugin/
It allows you as well to import posts with an add-on to wp-all-import plugin.
But this is for Import, not export.
Last but not least, have you seen https://toolset.com/download/toolset-advanced-export/?
It allows you to export various settings which are not covered by the WordPress XML export as seen in Tools > Export.
But in that you can no0t handle posts, I just wanted too share that in case you can use it.
You can export Reading settings, Customizer setup, Menu setup and Widget setup.
That is basically all we cover in Toolset.
The particular feature you mention would need something new, that is not within the scope of Toolset, but interesting.
I will keep this as a note, and eventually we can iterate such a feature in future, but for now this is not possible.
it would however be possible to display those posts with Views.
Do you need information how to do that?
