We would do the troubleshooting as below:
1. We would check the connector on the two email system, confirm then sender connector on the site and the Receive\Send connector configuration are proper.
2. Does the email flow work well for the mailbox which hosted on the exchange server, run EXBPA on the exchange server, confirm any information
3. Run DCDIAG and NETDIAG on the dc server, and on the exchange server, confirm whether there are some useful information
4. Open ADSI, launch below:
Configuration-> Services -> First Organization -> Administrative groups ->Servers -> Mailboxsever -> Properties -> Security
Confirm the exchange servers has the “Include Inheritable Permission” checked.