Say I have a trivial container based on the ubuntu:latest. Now there is a security update and ubuntu:latest is updated in the docker repo .
One of the ways to do it is to drive this through your CI/CD systems. Once your parent image is built, have something that scans your git repos for images using that parent. If found, you'd then send a pull request to bump to new versions of the image. The pull request, if all tests pass, would be merged and you'd have a new child image based on updated parent. An example of a tool that takes this approach can be found here: https://engineering.salesforce.com/open-sourcing-dockerfile-image-update-6400121c1a75 .
If you don't control your parent image, as would be the case if you are depending on the official ubuntu image, you can write some tooling that detects changes in the parent image tag or checksum(not the same thing, tags are mutable) and invoke children image builds accordingly.