This is a great question and something we've been trying to figure out ourselves. Historically, we were using Ansible to deploy Docker containers to EC2 instances, but have moved some services over to Kubernetes, Swarm and Lambda/Serverless. All of these are create the same deployment challenges -- the current products out there don't fit perfectly. The more we want to deploy to a higher level than "just Docker", the less Ansible provides today. But we wanted to stick to the core concepts of automation, continuous delivery (at least to staging), and chatops style management of production.
Our current approach is using an Operable (https://operable.io) Cog we wrote which takes the kubernetes yaml and applies it to a running cluster. It's not perfect, but I'm pretty happy with the direction it's going. We built this cog in a public repo (https://github.com/retracedhq/k8s-cog) so you are welcome to use any of it, if it's useful. Then we have our CI service send a message (using SQS) after a build is done to deploy to staging.
Our current approach is using an Operable (https://operable.io) Cog we wrote which takes the kubernetes yaml and applies it to a running cluster. It's not perfect, but I'm pretty happy with the direction it's going. We built this cog in a public repo (https://github.com/retracedhq/k8s-cog) so you are welcome to use any of it, if it's useful. Then we have our CI service send a message (using SQS) after a build is done to deploy to staging.