Hi, i've spend some time searching the way to export huge library from 2010 server and import it to 2013 server but have not found complete and free solution, so here is the way :
First use granular backup at source server, select whatever you want so we get cmp file or many cmp files ( depending on library/list/site size ). Then open it like cab, extract contents. If you got lot of files then extract them all and place all contents
to one folder. Edit file SystemData.xml and change version number from 14 to 15 like this :
<SchemaVersion Version="15.0.0.0" Build="15.0.6120.5006"
Save, pack all contents back to cab.
If multiply cabs was extracted pack them to one file anyway, no other changes needed ( so you will got one big cab))
Rename it back to cmp.
On destination server in management console write
Import-SPWeb http://sitepath -path c:\yourcab.cmp -updateversions overwrite
Thats all!
There is more completely described posts ( http://www.sharepoint-journey.com/backup-and-restore-sharepoint-2013-part1.html#gallery[pageGallery]/0/ for example ), but none of them describes how to migrate data with different server versions and non describes how to handle multiply cmp's so hope this help someone.
p.s.
After export your new list/lib or whatever will look like 2010 ( if you export to 2013 ), to fix this just edit it in SharePoint designer - delete existing view page and create new one.