Is there a way to copy SharePoint 2010 list to SharePoint 2013
Including list items and workflows (SharePoint designer workflows)
No approval for third party migration tool.
Unfortunately, unless you start writing some custom code you won’t be able to migrate the workflows over. However, are you looking at workflow history or just the actual workflows?
If your Workflows are Content Type workflows you could quite possibly re-deploy them to the new site if you recreated the workflows along with their GUIDs exactly. You can do this with Powershell and I actually just provided an answer to a question that illustrates how to do that here: Then when you recreate the content type, you should be able to export the data from 2010 to 2013 and then deploy the WFs by exporting from 2010 and importing them into 2013.
If the above doesn’t match your situation or you need the history, I know of no way OOTB that you can accomplish it. You will simply need to either export the data and recreate the WFs (without History) or write something that will move the data over along with workflows and history. However, the cost for to your business to do that, may be more than purchasing a tool to do it for you.
Hope this help,
Dave
