Sending them to a local secondary school may not result in them really being local. I know from our local primary that the boys seem to have a hierarchy of Harris, Kingsdale and Forest Hill boys of the locals with a few looking at grammars, church, private and some moving to different areas.
They may actually find more of their peers at Harris than if they went to FH boys. I guess the only truly local schools are those with small catchment area whereas FH has a big catchment area and your kids are more likely to sit in class with kids who have come to school by bus than walked.
There is no right or wrong just what is best for your kid. I do support the idea of local schools having been to the one and only local school where I lived but I guess choice has ruined that concept here.