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.
To enable this feature, go to Settings at the sidebar menu and enable the option Calculate bundle master inventory (every 10 minutes). It is successfully enabled when the colour is blueish like 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