Hi ,
1.What happens did you try to send an email to anyone of the user in exchange 2010 from exchange 2013 owa by logging in to the migrated mailbox ?
2.What happens when you to try to send an email form anyone of the newly created mailbox in exchange 2013 to anyone of the mailbox user in exchange 2010 ?
3.Did you have any transport rules in place ?
4.Does this issue happens for all the migrated mailboxes from exchange 2010 ?
If any of the above doesn't help then the final option would be to enable the transport pipeline logs to find out where the message is getting moderated .However first we need to test the basic things which is mentioned above.