Consider the scenario where you have a Summary Links Web Part (part of the SharePoint Publishing functionality) configured on a page and you want to import the preconfigured Web Part on a different page on a different site. If you try this you’ll get “List does not exist”:
Note that importing the Web Part in the same site (same or different pages) works just fine.
This is because the Summary Links Web Part references the list that contains the page where the Web Part resides on. If you open the .webpart file in a text editor you’ll see ListName and ListId containing the GUID of that list. So it can be the “Site Pages” library, the “Pages” library, or any Document Library that has Web Part Pages.
You can verify this by navigating to the following URL (note to replace the actual GUID): http://sitename/_layouts/listedit.aspx?List=GUID
Bonus question: what is the value when the Summary Links Web Part is on the “default.aspx” of a site? Answer:
So what’s the solution? Just remove the ListName and ListId elements (or their values) from the exported .webpart file and you’ll have no issues importing it to other sites.