Q. If I have ExpressRoute will Azure Site Recovery replication use the ExpressRoute connection?
A. There are two primary types of peering (connectivity) to Azure with ExpressRoute, private and public. Private connects on-premises to virtual networks in Azure while public connects on-premises to pretty much all of the other Azure services which includes ASR. Therefore if you have ExpressRoute and enable public peering the ASR replication traffic will leverage the ExpressRoute connection.