Below the best article, that I've been able to find regarding this solution.
http://msexchangeguru.com/2013/05/10/exchange2013-migration/
We've faced some problems by the migration of Exchange 2010 to Exchange 2013. By following the article above, you'll be able to get the migration done, but keep the following points in mind, in case you'll face any problems:
1. Both servers need to have exactly the same authentication settings enabled. This one solved a lot of problems for us. Please check or set following the settings:
[PS] C:\Windows\system32>Get-OutlookAnywhere | Set-OutlookAnywhere -IISAuthenticationMethods: Basic, ntlm
[PS] C:\Windows\system32>Get-OutlookAnywhere | fl
//this attributes must be exactly the same on both server 2010 and 2013 (check carefully)
ExternalClientAuthenticationMethod : Ntlm
InternalClientAuthenticationMethod : Ntlm
IISAuthenticationMethods : {Basic, Ntlm}
XropUrl :
ExternalClientsRequireSsl : True
InternalClientsRequireSsl : False
2. If you face any problems with the Activesync by Android mobile devices, then "Just disable password request in the mobile policy settings!"
3. If you face any problems with the move of the Arbitration mailboxes or the Rooms, then just follow the instructions from the Microsoft link:
https://technet.microsoft.com/en-us/library/dn249849(v=exchg.150).aspx
Check first what mailboxes are where: Get-Mailbox -Arbitration | FL Name,DisplayName,ServerName,Database,AdminDisplayVersion and Get-mailbox -server <Exch2010>| where{$_.recipienttype -like "*room*"}
Move the mailboxes: Get-Mailbox -Arbitration -Identity "SystemMailbox{e0dc1c29-89c3-4034-b678-e6c29d823ed9}" | New-MoveRequest -TargetDatabase "exchange 2013 database"
**The rest should work without problems by just following the steps in the exchange guru article (incl. the links in the article).