I know of no such rule. Some people, hidebound by etymology (Latin cum) go out of their way to use 'with' in every case; the same people will say 'in the circumstances' rather than 'under the circumstances'.
'General run' here is using 'run' as it is used in 'run-of-the-mill' - meaning ordinary. (This has nothing to do with an Access-All-Areas pass to a windmill. It is the sort of mill associated with mass-production.)