Some messages in target mailbox have an Internet Header, but some don't have one.
Is there something wrong?
There is nothing wrong, not all messages have the Internet Header populated.
Usually a message which has been sent within the same organization doesn't have any information under Internet Header, and only messages which have been sent trough Internet will have the Internet Header populated.
QMM Exch by default copies the entire message with all message attributes, including the Internet Header (if present).
Every single e-mail you received from other organizations on the Internet contains an Internet Header. A valid Internet e-mail header provides a detailed log of the network path taken by the message between the mail sender and the mail receiver(s), this Internet mail header can sometimes be quite long depending on the network path between sender and receiver.