Depends. If your marketplace shows out of stock inventory, it will appear. If you do not show out of stock inventory, it will not appear. You can limit via API not promoting out of stock products.
Our dashboard for ad creation is dynamic and responsive in real time. You will not be able to create a campaign with out of stock inventory. If a campaign is running and a product runs out of stock, the out of stock product will not be entered into any more auctions.