Hi there,
I would recommend making sure that your Shopify store is configured to prevent customers from purchasing out of stock products. As explained in our Help Center article, “Klaviyo will only "un-publish" out-of-stock items for Shopify stores if you do not allow customers to purchase products when they go out-of-stock. If you have multiple variations of an item, all variations must be out-of-stock in order for Klaviyo to consider the entire item out-of-stock and unpublish it.”
Once you’ve made sure that you do not allow customers to purchase products when they out of stock, you can follow the instructions in the linked article to prevent “unpublished” products from being displayed in emails. Since Klaviyo will then automatically un-publish out of stock products, out-of-stock products will not be displayed in emails as long as you update your product blocks according to the instructions included in the article.
Best,
Caroline