How to manage theme, plugin and core changes in a Dockerised WordPress instance?

Solution:

Typically you do not update the data within a container dynamically. The Docker image is built as a complete image with all its dependencies – server software, application code etc.

The typical workflow for ECS is to push your containers to the Elastic Container Registry (ECR) and then update your Task Definition to use that new image. This triggers the containers to update with the new image.