Ask HN: How do you do Docker CI?

I have a bunch of Docker containers all hooked up using docker-compose (previously fig). I have found that services that advertise themselves as "Docker CI" services are usually just talking about using Docker to run their build workers.

I would like a continuous integration service that preserves the Docker build cache between runs, instead of spinning up a completely new worker/filesystem/build context. Otherwise, builds take absolutely forever. Ideally managed (unlike https://github.com/groupon/DotCi, which doesn't come with CircleCI's ready-to-go per-build database).

The only thing I've found is TeamCity (old, you can't version build tasks in a yaml file easily like you can with CircleCI, you have to maintain infrastructure, etc). Quay.io doesn't provision a database like CircleCI and Travis do, so you have to bifurcate dev and prod and install one inside the container and run unit tests during the container build (ew - you can't spawn into an unbuilt container to debug things!).

How do you run continuous integration on your Docker container cluster?