The key word that covers the Heroku-style experience is "GitOps".

Most tools are Kubernetes-centric (i.e. FluxCD[0] or Jenkins X[1]), but there are some simpler Docker-only options as well such as Dokku[2].

[0]: https://fluxcd.io/

[1]: https://jenkins-x.io/

[2]: http://dokku.viewdocs.io/dokku/