If for whatever reason you have to have Domain Controllers separated by firewalls you will require the following ports and protocols allowed between the DC's to ensure you can A.) join the domain and B.) maintain a healthy Active Directory with replication etc.
LDAP TCP-in - 389
LDAP UDP in - 389
LDAP for Global Catalog TCP in - 3268
NetBIOS name Resolution UDP in - 138
SAM/LSA TCP in - 445
SAM/LSA UDP in - 445
Secure LDAP TCP in - 636
Secure LDAP for Global Catalog TCP in - 3269
W32Time NTP UDP in - 123
RPC - RPC Dynamic
RPC Endpoint Mapper
DNS - TCP and UDP 53
Kerberos V5 UDP in - 88
Netbios Datagram UDP in - 137
There are multiple ways to have Active Directory extended to none-trusted networks. Since Windows Server 2008 R2 the option to have a Read Only Domain Controller (RODC) in an un-trusted network has proved attractive to business. This ensure that if the un-trusted network is compromised, the RODC only holds a replicated copy of the Active Directory database. Even if the RODC was taken offline and attacked, no "write back" to Active Directory is possible so the impact is minimal.
Depending on your requirements options such as Active Directory Federation Services (AD FS), DirSync, FIM 2010 R2 and Microsoft Identity Manager 2015 (MIM) can all be good solutions for identity and access management in an Active Directory environment.