The Products Kits App listens to the Webhooks fired by Shopify to sync up the product inventory.
Shopify does not fire any Webhook after a transfer takes place. In other words, the Product Kits app does not get any notification if the inventory is updated via transfer.
So, in this case, you will be required to manually sync the inventory. Here are the instructions
Products Inventory Update In The App After Transfer:
To update the product inventory in the app after transfer, you have to manually sync the bundle. It will pull the latest inventory values in the app. To manually sync the bundle, simply go to the ‘Help & Support’ -> ‘Manually Sync Bundles’.
Here is the screenshot.