Note: We’ve renamed our SmartConnectors to Integration Apps.
The kit inventory saved search uses a summary filter hardcoded in the search itself, which allows the search to directly retrieve only those kits whose member items inventory have changed within the last 'X' minutes. Currently it is hardcoded to 30 minutes, which means that the search will only show those kits whose member items inventory have changed within the last 30 minutes. Check the filter under the Saved Search -> Filters tab-> Summary Filter subtab, and change 30 minutes to a different time, if needed. Ideally this time should be the same as the flow schedule.
The reason the original search was not producing any results could have been because there were no kits whose inventory had changed within the last 30 minutes.