Discount Ninja uses standard Shopify charges.
Therefore the documentation authored by Shopify applies: https://help.shopify.com/en/manual/your-account/manage-billing/your-invoice/apps#app-usage-charges
For your convenience here's the most relevant section from that article, authored by Shopify:
App prorating, upgrades, and downgrades
If you upgrade or downgrade your subscription with an app, then the app will prompt you to agree to a new recurring app charge. This is because Shopify allows each app only one recurring app charge to be enabled at a time. The existing recurring application charge will be canceled and replaced by the new charge.
When you upgrade your plan by moving from a less expensive charge to a more expensive charge, the charge is prorated based on the difference in price and the number of days remaining in the billing cycle. For example, if you begin a 30-day billing cycle on a $5.00 plan, and then upgrade to a $15.00 plan on day 15 of the billing cycle, you'd be charged $5.00 + ($15.00 - $5.00) * (15/30) = $10.00 USD.
If you downgrade your plan by moving from a more expensive charge to a less expensive charge, then you're automatically offered an application credit based on the difference in price and the number of days remaining in the billing cycle. This application credit can be used towards any future application purchase on Shopify.