The Site Activity web part is a dynamic web part that automatically displays activity from you site. This activity can include edits, file uploads, creation of list and libraries, and so on.
You run a copy structure on your site, and the Site Activity web part at the destination does not display any content other than a welcome message.
ShareGate Desktop cannot migrate the content of a Site Activity web part from a source site to a destination site. When the app migrates any web part, it verifies that the data at the source is also available at the destination. In the case of a dynamic web part such as this one, that is not possible. The content is updated live as activity occurs on the source site. This data will not exist at the destination.
There is no way to retrieve the site activity from your source site, but the Site Activity web part on your destination site will automatically begin displaying new content as it occurs at the destination.