Hi @hsipani,
When a product that has previously been added to a customers cart is now out of stock if the customers visit the cart page they will see this message below displaying on the page. The customer cannot then purchase the product that is in the cart.

Image Link: https://cloudup.com/cAhl9i-6c9w
Below is an example of what would happen if they did try to proceed to checkout.

Image Link: https://cloudup.com/cV5my0B3KJ3
I’ve performed some cross-testing for us today on several of my sites which have the latest versions of WordPress, WooCommerce and our free Storefront theme as the only enabled software. When doing so I received the same results on all sites I tested with.
Below is a screenshot of how I tested.

Image Link: https://cloudup.com/chQ0lsT9ljS
If this is not working for you the reason wouldn’t be directly related to the WooCommerce plugin itself. It could be a conflict from a plugin or theme installed and activated on your WordPress installation.
I hope this helps.