How does Amazon Private Label handle out-of-stock situations for products?

The strategies to handle such situation may include the following:
Automatic Replenishment, Backorders, Pre-Orders, Alternative Products and Disabling the Listing.

it’s important to have a solid inventory management strategy in place to avoid running out of stock on Amazon. By using Amazon’s inventory management tools, automated replenishment system, and backorder options, you can help minimize the impact of out-of-stock situations on your sales and reputation.

You need to manage your inventory by Automatic Replenishment or some other means and you can also use the technique of Copying the same Listing as FBM so that whenever you go out of stock you can maintain the Ranking by selling as FBM.