No, scheduled searches are not that smart. It will run over 'All Time' every time it runs, that is the time-range that you have originally configured.
I realise that this is only an example, but in general, 'all time' searches are very resource intensive and scheduled searches should be optimized as much as possible. If you want to run a search every 24 hours, then you only need to search the last 24 hours - starthoursago=24 - similarly, if you're running an hourly search, you'll want to specify something like starthoursago=1
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