Hi Felix,
Sorry for the misunderstanding of your situation previously. I’d like to share some points about moving the contents from Office 365 P1 plan to Office 365 E3 plan with you in order to help achieve the task.
Regarding the E-mails contents, to make sure the process will be handled smoothly, I suggest you post the requirement under the sub-category of the forum: Email and calendar, and our engineers will focus on your query and endeavor to assist you in the new thread.
For the SharePoint Online contents, as we know, in P1 plan we have Team Site (including its subsites) and public website, however, in E3 plan we can create several site collections from SharePoint admin center (Admin>SharePoint>SharePoint admin center>site collections).
To move the contents (such as library/list), we can save the library/list as template from P1 plan (LIBRARY>Library Settings>Save the document library as template, Include Content), and download it (.stp file) from List templates (upper right gear Settings>Site settings>Web Designer Galleries>List templates). And then upload the library/list file (.stp file) to List templates in E3 plan to use it. I understand that it could take some time to finish moving all contents.
Moreover, I also notice that “I tried to export Lists as content but I always get a message about missing features. I already checked the feature lists of both sites but they are exactly the same” you said. I’d like to confirm where and how you perform the action. Could you please providing the steps you tried so we can reproduce the error? It would be appreciated if you could capture a screenshot of the message and post it back. Please upload the screenshot by clicking Use rich formatting and selecting Insert Image.
Thanks,
Mandy Meng