I did my upgrade some site collection/s at the time, moving them in separate databases. I answered a similar question a year ago so i'm going to quote myself from that thread.
While the official way to do it provided by Microsoft is to Migrate from classic-mode to claims-based authentication by converting a web application.
"However, i was planning to move site collections in a temporerary
database, one at the time for several reasons. I searched and found this script
that simply converts a database from Classic to Claims. I've used it
for multiple databases and has not had one single error related to
authentication in 2013.
The script is intended to be run AFTER the database has been mounted
to the web application. ( mount-spcontentdatabse ). This script must
be executed by your Farm account to. Check the source for the script
for more information, and be sure to test it before using it in
production if you decide to use it."
$wa = Get-SPWebApplication http://yourWebAppUrl
$acc = 'domain\user'
$arguments = New-Object Microsoft.SharePoint.Administration.SPWebApplication+SPMigrateUserParameters
$wa.MigrateUsersToClaims($acc, $true, $arguments)
So, move a site collection from 2010 to its own temporary content database, move it a 2013 environment, mount it to a web application and then run the script. The important thing is that you have to target the script to the right database so if you have a new web application with one database and then attach a temporary content database, the database added order will be
$wa.ContentDatabases. If you have a web application with 3 databases and you mount a fourth database, the number should be
$wa.ContentDatabases as you start to count from 0.
I did this for all of our site collections in the spring -14 and we have not had any problems related to authentication after our migration.
Migrate from classic-mode to claims-based authentication in SharePoint 2013 the easy way
Old thread: web application is configured with claims authentication mode however the content database you are trying to attach is intended