Hi kelsnz,
As I understand, the task lists don't aggregate to My Sites in a migrated site collection or a new one in SharePoint 2013.
You could check things below:
1. Check if the web application pool account you used for My Sites has SPDataAccess SQL permissions in your other Content databases of your web applications. If your account does not have the permission, the marking tasks as completed will not work.
After checking, you could restart the Work Management Service Application.
2. Check if the Work Management Service Application use the same account as the web application.
3. Check if the service account used for the Work Management Service Application has full control permissions on the User Profile Service.
You could follow the steps to add the full control permission on the user profile service:
1. Under Application Management section in Central Administration, click Manage service applications.
2. In the list of service applications, click User Profile Service Application.
3. On the Service Applications tab, in the Operations section, click Administrators.
4. On the Administrators for User Profile Service Application page, type the service account of the WMA and then click Add.
5. In the Permissions for Administrator box, check the Full Control permission level, and then click OK.
5. Check if the Task content type is marked as the Default Content Type for your custom list in order for the tasks to be aggregated by the service.
6. Check if the web application is associated with the application proxy group which hosts the WMA service application.
The article below is about Work Management Service Application.
http://blogs.technet.com/b/praveenh/archive/2013/08/22/work-management-service-application.aspx
Best regards,
Sara Fan