What is the behavior of the BIG-IP DNS monitor following datacenter synchronization?

In theory, the VS doesnt really perform health monitoring. The BIG-IP does. The way this is supposed to work with prober pools, is that you want to use the local datacenter resources to monitor those origin servers which are in the same datacenter. Of course, when you create a prober pool, you must also select a datacenter. So, a resource within the same datacenter, would use a local prober pool which is assigned to the same datecenter, unless you have it hard coded otherwise. The BIG-IP in the other datecenter will receive the monitor status and metrics using BIG3D and iQuery. However, if a BIG-IP is not available, then the surviving BIG-IP will perform health monitoring directly toward the other datacenter.

Have a look here for some additional information: https://techdocs.f5.com/kb/en-us/products/big-ip-dns/manuals/product/bigip-dns-implementations-13-0-0/17.html.