F5 Local traffic Pool -> How does it track the availability of the members of the pool?

The F5 uses the IP closest to the destination as the IP for health monitoring. So if you had 3 interface excluding management, external, dmz, and internal and your destination IP for a pool member existed in a subnet routed out the external interface you would see the health monitor come from the external interface, this is regardless of which subnet the virtual server exists in that the pool in question is configured under. You could theoretically have pool members in those 3 segments and balance traffic to them but that would cause a whole other issue with needing to configure SNAT but that's a different question. Even in an HA configuration the health monitor comes from the self-ip of the closest interface. I'm not aware of any way to force health monitors to come from a specific IP other than having that IP be the interface IP of the F5 in question.