Connected Inventory can update the Bundle master product's inventory quantity automatically. The app updates the Bundle Master's inventory based on how many Component are left for every 10 minutes.
How to enable 10 minutes auto calculation?
Select Bundle Groups at the sidebar menu and check the Bundle master calculation settings. If Bundle master calculation settings is enabled, 10 minutes auto calculation is activated.
If not, click the enable button to activate 10 minutes auto calculation as per the screenshot below.
Note that even when inventory quantity is down to zero, it is up to your product setting to decide whether it would display an Out Of Stock message to your customer. Refer to this other FAQ: Why the product does not display sold out