How to test previous version of application with new database version?

Let's say now I have my app deployed to production - v1 and it uses a database with version v1. Now I want to deploy a new version of app v2 with database version v2.

I suppose that database changes are backward-compatible, and the v1 app can work with v2 database. But I want to be sure that it works by running tests for the app v1 with database v2.

Is there any ready solutions/techniques for this?

Currently I have an automated pipeline before deploy that works like this:

1. Run a new database container
2. Populate database, do other related stuff...
3. Run tests

But I don't have ideas how to automate previous app version testing.